A Dissertation Entitled Toward Auxetic Composites for Structural

A Dissertation

entitled

Toward Auxetic Composites for Structural Applications: Finite Element Analysis of

Origami-Inspired Foldcores

Ehsan Nabiyouni

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the

Doctor of Philosophy Degree in Mechanical Engineering

______Dr. Lesley Berhan, Committee Chair ______Dr. Sarit Bhaduri, Committee Member ______Dr. Maria Coleman, Committee Member ______Dr. Mohammad Elahinia, Committee Member ______Dr. Brian Trease, Committee Member ______Dr. Amanda Bryant-Friedrich, Dean College of Graduate Studies

The University of Toledo

May 2017

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

An Abstract of

Toward Auxetic Composites for Structural Applications: Finite Element Analysis of

Origami-Inspired Foldcores

Ehsan Nabiyouni

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the PhD Degree in Mechanical Engineering

The University of Toledo

May 2017

Sandwich panels are used in several industries. A sandwich structure is made of a light-weight, lower strength core sandwiched between two thin, stiff facesheets. The combination of light weight and desired mechanical properties makes these structures attractive for construction, automotive, aerospace and many other industries.

Conventional honeycombs, which have high stiffness and are lightweight are widely used as the cores of sandwich structures.

The main objective of this study was to investigate the behavior of three structures with negative effective Poisson’s ratios and one widely used geometry as the

cores of sandwich panels and compare their performance under different mechanical loading conditions such as compression, shear, bending and impact.

A total of 294 simulations were done, and the results showed different behaviors of the geometries under the compressive loads in z, x and y directions as well as shear loads in x and y directions. Regular and inverted honeycombs demonstrated higher effective Young’s moduli under compressions in z direction as well as shear loads whereas the foldcores were dominant in compressive loads in x and y directions. When facesheets were added, all models showed enhancement in effective Young’s moduli in x and y directions while the one for z direction and also shear moduli did not remarkably change. In addition, there were different behaviors of these cores and sandwich structures under bending and impact loads. As expected, inverted honeycomb formed a dome- shaped structure under bending. But, surprisingly, waterbomb created an anticlastic (i.e. dome-shaped) structure under bending as well. In the case of impact, inverted honeycomb combined by with two facesheets absorbed about 85% of impact energy and was superior to other models.

For all simulations, comparisons based on geometry and loading were performed, and presented in tables and figures, thus, one can pick a geometry which is more suitable for their applications.

Acknowledgements

I would like to express my appreciation to my advisor, Dr. Lesley Berhan, for her enthusiasm, patience, motivation, and knowledge. Also, I am grateful for other members of my dissertation committee, Dr. Mohammad Elahinia, Dr. Brian Trease, Dr. Maria

Coleman and Dr. Sarit Bhaduri for their supervision.

I am grateful for the chair of MIME department, Dr. A. Afjeh for supporting me during this study, and also, I would like to appreciate his effort and generosity in providing necessary simulation packages for this work.

I would like to indicate gratitude for Dr. Nariman Mansouri for his support throughout this study. Also, I am thankful for all of my friends for their advice and help.

Finally, I am most appreciative to my family especially my parents for their love and support throughout my life.

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Contents Acknowledgements ...... iii List Tables ...... vii List of Figures ...... x 1. Introduction ...... 1 2. Literature Review ...... 5 3. Motivation and Objectives, and Procedure ...... 23 3.2. Procedure ...... 24 3.2.1. Material ...... 26 3.2.2. Geometry ...... 26 3.2.3. Loading ...... 30 3.2.3.1. Compression in the z direction ...... 30 3.2.3.1.1. Inverted honeycomb ...... 30 3.2.3.1.2. Waterbomb ...... 32 3.2.3.1.3. Regular honeycomb ...... 33 3.2.3.1.4. Chevron ...... 34 3.2.3.2. Compression in the x direction ...... 36 3.2.3.2.1. Inverted Honeycomb...... 37 3.2.3.2.2 Waterbomb ...... 37 3.2.3.2.3 Regular honeycomb ...... 37 3.2.3.2.2 Chevron ...... 37 3.2.3.3. Compression in the y direction ...... 37 3.2.3.3.1. Inverted honeycomb ...... 37 3.2.3.3.2. Waterbomb ...... 38 3.2.3.3.3. Regular honeycomb ...... 38 3.2.3.3.4. Chevron ...... 38 3.2.3.4. Shear in the x direction ...... 38 3.2.3.4.1 Inverted honeycomb ...... 38 3.2.3.4.2 Waterbomb ...... 39 3.2.3.4.3 Regular honeycomb ...... 39 3.2.3.4.4 Chevron ...... 39 3.2.3.5. Shear in the y direction ...... 39 3.2.3.5.1. Inverted honeycomb ...... 39 3.2.3.5.2. Waterbomb ...... 40 3.2.3.5.3 Regular honeycomb ...... 40 3.2.3.5.4 Chevron ...... 40 4. Compression and Shear for the Cores ...... 41 4.1. Results ...... 41 4.1.1. Compression in the z direction ...... 41 4.1.1.1. Inverted honeycomb ...... 41 4.1.1.2 Waterbomb ...... 42 4.1.1.3. Regular honeycomb ...... 43 4.1.1.4. Chevron ...... 44 4.1.2. Compression in the x direction ...... 45 4.1.2.1. Inverted honeycomb ...... 46 4.1.2.2. Waterbomb ...... 47 iv

4.1.2.3. Regular honeycomb ...... 48 4.1.2.4. Chevron ...... 49 4.1.3. Compression in the y direction ...... 49 4.1.3.1. Inverted honeycomb ...... 50 4.1.3.2. Waterbomb ...... 50 4.1.3.3. Regular honeycomb ...... 51 4.1.3.4. Chevron ...... 52 4.1.4. Shear in the x direction ...... 53 4.1.4.1. Inverted honeycomb ...... 54 4.1.4.2. Waterbomb ...... 54 4.1.4.3. Regular honeycomb ...... 55 4.1.4.4. Chevron ...... 56 4.1.5. Shear in the y direction ...... 57 4.1.5.1. Inverted honeycomb ...... 58 4.1.5.2. Waterbomb ...... 59 4.1.5.3. Regular honeycomb ...... 59 4.1.5.4. Chevron ...... 60 4.2. Analysis and comparison ...... 61 4.2.1. Geometry ...... 64 4.2.2. Loading Conditions ...... 72 5. Compression and Shear for Sandwich Panels ...... 82 5.1. Results ...... 83 5.1.1. Compression in the z direction ...... 83 4.1.2. Compression in the x direction ...... 91 4.1.3. Compression in the y direction ...... 99 4.1.4. Shear in the x direction ...... 107 4.1.5. Shear in the y direction ...... 115 5.2. Analysis and comparison ...... 123 5.2.1. Geometry ...... 123 5.2.2. Loading Conditions ...... 142 6. Curvature for the Cores ...... 153 6.l. Results ...... 153 6.1.1. Distributed edge load ...... 153 6.1.1.1. Inverted honeycomb ...... 153 6.1.1.2. Waterbomb ...... 156 6.1.1.3. Regular honeycomb ...... 158 6.1.1.4. Chevron ...... 160 6.1.2. Concentrated center load ...... 162 6.1.2.1. Inverted honeycomb ...... 162 6.1.2.2. Waterbomb ...... 164 6.1.2.3. Regular honeycomb ...... 166 6.1.2.4. Chevron ...... 168 6.2. Analysis and comparison ...... 170 6.2.1. Distributed load ...... 170

6.2.2. Concentrated load ...... 173 7. Impact ...... 177 7.1. Results ...... 179 7.2. Analysis and Comparison ...... 200 8. Conclusions and Future Work ...... 208 References ...... 221

List Tables

Table 3-1: Material’s properties ...... 26 Table 4.1: Values used for normalizing the results ...... 63 Table 4.2: Inverted honeycomb’s normalized effective Young’s modulus under different compression loading conditions ...... 64 Table 4.3: Inverted honeycomb’s normalized effective shear modulus under different shear loading conditions ...... 64 Table 4.4: Waterbomb’s normalized effective Young’s modulus under different compression loading conditions ...... 67 Table 4.5: Waterbomb’s normalize effective shear modulus under different compression loading conditions ...... 67 Table 4.6: Regular honeycomb’s normalized effective Young’s modulus under different compression loading conditions ...... 69 Table 4.7: Regular honeycomb’s normalized effective shear modulus under different shear loading conditions ...... 69 Table 4.8: Chevron’s normalized effective Young’s modulus under different compression loading conditions ...... 70 Table 4.9: Chevron’s normalize effective shear modulus under different compression loading conditions ...... 71 Table 4.10: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the z direction...... 73 Table 4.11: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the x direction...... 75 Table 4.12: Different models’ normalized effective Young’s modulus and Poisson’s ratios under compression in the y direction...... 76 Table 4.13: Different models’ normalized effective shear modulus shear in the x direction ...... 78 Table 4.14: Different models’ normalized effective shear modulus shear in the y direction ...... 78 Table 5.1: Mechanical properties of CFRP ...... 82 Table 5.2: Mechanical properties of adhesives ...... 83 Table 5.1: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 124 Table 5.2: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 124

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Table 5.3: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 125 Table 5.4: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 125 Table 5.5: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 128 Table 5.6: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 129 Table 5.7: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions..... 129 Table 5.8: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 130 Table 5.9: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 133 Table 5.10: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 133 Table 5.11: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 134 Table 5.12: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 134 Table 5.13: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 137 Table 5.14: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 138 Table 5.15: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 138 Table 5.16: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 139 Table 5.17: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the z direction...... 142 Table 5.18: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the x direction...... 145 Table 5.19: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the y direction...... 147 Table 5.20: Different models’ normalized effective shear modulus shear in the x direction ..... 149

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Table 5.21: Different models’ normalized effective shear modulus shear in the y direction ..... 150 Table 5.22: The best two geometries for each loading ...... 152 Table 6.1: Radius of curvatures for inverted honeycomb under distributed edge load ...... 170 Table 6.2: Radius of curvatures for waterbomb under distributed edge load ...... 171 Table 6.3: Radius of curvatures for regular honeycomb under distributed edge load ...... 171 Table 6.4: Radius of curvatures for chevron under distributed edge load ...... 172 Table 6.5: Radius of curvatures for inverted honeycomb under concentrated center load...... 174 Table 6.6: Radius of curvatures for waterbomb under concentrated center load ...... 174 Table 6.7: Radius of curvatures for regular honeycomb under concentrated center load ...... 175 Table 6.8: Radius of curvatures for chevron under concentrated center load ...... 175 Table 7.1: maximum stress in the cores...... 183 Table 7.2: Percentage of energy absorbed by the model under 10 mJ impact...... 186 Table 7.3: Percentage of energy absorbed by the model under 20 mJ impact...... 190 Table 7.4: Percentage of energy absorbed by the model under 30 mJ impact...... 193 Table 7.5: Percentage of energy absorbed by the model under 40 mJ impact...... 196 Table 7.6: Percentage of energy absorbed by the model under 50 mJ impact...... 200 Table 7.7: All results for impact ...... 200

List of Figures

Figure 1-1: Honeycomb core structure [2] ...... 2 Figure 1-2: Chevron foldcore structure [3]...... 2 Figure 1-3: Modified Chevron foldcore structure [3]...... 3 Figure 1-4: Flat-folded (right) and curved (left) Waterbomb [3] ...... 4 Figure 2-1: Ideal cellular structure of reentrant honeycombs [2] ...... 6 Figure 2-2: Schematic of inverted honeycomb showing transverse expansion when stretched (top) and (1, 4)-refexyne with a negative Poisson’s ratio (bottom) [2]...... 7 Figure 2-3: Material’s density below the impact point is different in auxetic and non-auxetic materials [2]...... 8 Figure 2-4: Schematic of conventional honeycomb cores ...... 16 Figure 2-5: Schematic of corrugated cores ...... 17 Figure 2-6: Schematic of foam cores ...... 18 Figure 2-7: Schematic of woven cores ...... 18 Figure 2-8: Schematic of a foldcore ...... 19 Figure 3-4: Waterbomb generated in Rhinoceros [64] ...... 27 Figure 3-5: Imported Waterbomb to Comsol ...... 28 Figure 3-6: Inverted honeycomb generated in Comsol ...... 28 Figure 3-7: Imported Chevron to Comsol ...... 29 Figure 3-8: Regular honeycomb generated in Comsol ...... 30 Figure 3-9: Inverted honeycomb in Comsol Multiphysics [65] ...... 32 Figure 3-10: Waterbomb in Comsol Multiphysics [65] ...... 33 Figure 3-10: Regular honeycomb in Comsol Multiphysics [65] ...... 34 Figure 3-12: Chevron in Comsol Multiphysics [65] ...... 36 Figure 4-1: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the z direction...... 42 Figure 4-2: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the z direction...... 43 Figure 4-3: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the z direction...... 44 Figure 4-4: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Chevron model under 5% compressive strain in the z direction...... 45 Figure 4-5: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the x direction...... 46 x

Figure 4-6: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the x direction...... 47 Figure 4-7: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the x direction...... 48 Figure 4-8: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% compressive strain in the x direction...... 49 Figure 4-9: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the y direction...... 50 Figure 4-10: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the y direction...... 51 Figure 4-11: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the y direction...... 52 Figure 4-12: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% compressive strain in the y direction...... 53 Figure 4-13: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the x direction...... 54 Figure 4-14: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% shear strain in the x direction...... 55 Figure 4-15: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the x direction...... 56 Figure 4-16: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% shear strain in the x direction...... 57 Figure 4-17: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the y direction...... 58 Figure 4-18: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% shear strain in the y direction...... 59 Figure 4-19: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% shear strain in the y direction...... 60 Figure 4-20: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% shear strain in the y direction...... 61 Figure 4-21: Inverted honeycomb’s normalized effective Young’s modulus under different compression loading conditions ...... 65 Figure 4-22: Inverted honeycomb’s normalized effective shear modulus under different shear loading conditions ...... 66 Figure 4-23: Waterbomb’s normalize effective Young’s modulus under different compression loading conditions ...... 67 Figure 4-24: Waterbomb’s normalize effective shear modulus under different shear loading conditions ...... 68 Figure 4-25: Regular honeycomb’s normalized effective Young’s modulus under different compression loading conditions ...... 69

Figure 4-26: Regular honeycomb’s normalized effective shear modulus under different shear loading conditions ...... 70 Figure 4-27: Chevron’s normalize effective Young’s modulus under different compression loading conditions ...... 71 Figure 4-28: Chevron’s normalize effective shear modulus under different shear loading conditions ...... 72 Figure 4-29: Normalized effective Young’s modulus for all models for compression in the z direction...... 73 Figure 4-30: Effective Poisson’s ratios xz and yz for all models for compression in the z direction...... 74 Figure 4-31: Normalized effective Young’s modulus for all models for compression in the x direction...... 75 Figure 4-32: Effective Poisson’s ratios zx and zy for all models for compression in the x direction...... 76 Figure 4-33: Normalized effective Young’s modulus for all models for compression in the y direction...... 77 Figure 4-34: Effective Poisson’s ratios zy and zx for all models for compression in the x direction...... 77 Figure 4-35: Normalized effective shear modulus of all models for shear in the x direction...... 79 Figure 4-36: Normalized effective shear modulus of all models for shear in the y direction...... 79 Figure 5-1: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction...... 84 Figure 5-2: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction...... 85 Figure 5-3: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction...... 86 Figure 5-4: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction...... 87 Figure 5-5: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction...... 88 Figure 5-6: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction...... 89 Figure 5-7: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction...... 90

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Figure 5-8: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction...... 91 Figure 5-9: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction...... 92 Figure 5-10: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction...... 93 Figure 5-11: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction...... 94 Figure 5-12: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction...... 95 Figure 5-13: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction...... 96 Figure 5-14: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction...... 97 Figure 5-15: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction...... 98 Figure 5-16: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction...... 99 Figure 5-17: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction...... 100 Figure 5-18: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction...... 101 Figure 5-19: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction...... 102 Figure 5-20: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction...... 103 Figure 5-21: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction...... 104

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Figure 5-22: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction...... 105 Figure 5-23: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction...... 106 Figure 5-24: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction...... 107 Figure 5-25: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction...... 108 Figure 5-26: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction...... 109 Figure 5-27: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction...... 110 Figure 5-28: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction...... 111 Figure 5-29: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction...... 112 Figure 5-30: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction...... 113 Figure 5-31: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction...... 114 Figure 5-32: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 5 % shear strain in the x direction...... 115 Figure 5-33: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction...... 116 Figure 5-34: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction...... 117 Figure 5-35: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction...... 118

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Figure 5-36: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction...... 119 Figure 5-37: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction...... 120 Figure 5-38: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction...... 121 Figure 5-39: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction...... 122 Figure 5-40: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction...... 123 Figure5-41: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 126 Figure 5-42: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 126 Figure5-43: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 127 Figure 5-44: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 127 Figure5-45: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 130 Figure 5-46: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 131 Figure5-47: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions..... 131 Figure 5-48: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 132 Figure5-49: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 135 Figure 5-50: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 135

Figure 5-51: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 136 Figure 5-52: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 136 Figure5-53: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 139 Figure 5-54: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions ...... 140 Figure 5-55: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 140 Figure 5-56: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions ...... 141 Figure 5-57: Normalized effective Young’s modulus for all models for compression in the z direction...... 143 Figure 5-58: Effective Poisson’s ratios xz and yz for all models for compression in the z direction...... 143 Figure 5-59: Normalized effective Young’s modulus for all models for compression in the x direction...... 146 Figure 5-60: Effective Poisson’s ratios zx and yx for all models for compression in the x direction...... 146 Figure 5-61: Normalized effective Young’s modulus for all models for compression in the y direction...... 148 Figure 5-62: Effective Poisson’s ratios xy and zy for all models for compression in the z direction...... 148 Figure 5-63: Normalized effective shear modulus of all models for shear in the x direction. .... 151 Figure 5-64: Normalized effective shear modulus of all models for shear in the y direction. .... 151 Figure 6-1: load is applied to the edges parallel to the x direction ...... 154 Figure 6-2: xyz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load...... 154 Figure 6-3: yz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load...... 155 Figure 6-4: zx view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load...... 155 Figure 6-5: xyz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load...... 156 Figure 6-6: yz view of deformation, curvature and stress distribution of waterbomb under distributed edge load...... 157 Figure 6-7: zx view of deformation, curvature and stress distribution of waterbomb under distributed edge load...... 157

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Figure 6-8: xyz view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load...... 158 Figure 6-9: yz view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load...... 159 Figure 6-10: zx view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load...... 159 Figure 6-11: xyz view of deformation, curvature and stress distribution of chevron under distributed edge load...... 160 Figure 6-12: yz view of deformation, curvature and stress distribution of chevron under distributed edge load...... 161 Figure 6-13: zx view of deformation, curvature and stress distribution of chevron under distributed edge load...... 161 Figure 6-14: xyz view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load...... 162 Figure 6-15: yz view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load...... 163 Figure 6-16: zx view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load...... 163 Figure 6-17: xyz view of deformation, curvature and stress distribution of waterbomb under concentrated center load...... 164 Figure 6-18: yz view of deformation, curvature and stress distribution of waterbomb under concentrated center load...... 165 Figure 6-19: zx view of deformation, curvature and stress distribution of waterbomb under concentrated center load...... 165 Figure 6-20: xyz view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load...... 166 Figure 6-21: yz view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load...... 167 Figure 6-22: zx view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load...... 167 Figure 6-23: xyz view of deformation, curvature and stress distribution of chevron under concentrated center load...... 168 Figure 6-24: yz view of deformation, curvature and stress distribution of chevron under concentrated center load...... 169 Figure 6-25: zx view of deformation, curvature and stress distribution of chevron under concentrated center load...... 169 Figure 7-1: Absorbed energy by top plate under 10 mJ impact ...... 180 Figure 7-2: Stress distribution and deformation of inverted honeycomb under 10 mJ impact .. 181 Figure 7-3: Stress distribution and deformation of waterbomb under 10 mJ impact ...... 181 Figure 7-4: Stress distribution and deformation of regular honeycomb under 10 mJ impact .... 182 Figure 7-5: Stress distribution and deformation of chevron under 10 mJ impact ...... 182

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Figure 7-6: Stress distribution and deformation of upside down waterbomb under 10 mJ impact ...... 183 Figure 7-7: Absorbed energy by inverted honeycomb under 10 mJ impact ...... 184 Figure 7-8: Absorbed energy by waterbomb under 10 mJ impact ...... 185 Figure 7-9: Absorbed energy by waterbomb upside down under 10 mJ impact ...... 185 Figure 7-10: Absorbed energy by regular honeycomb under 10 mJ impact ...... 186 Figure 7-11: Absorbed energy by chevron under 10 mJ impact ...... 186 Figure 7-12: Absorbed energy by inverted honeycomb under 20 mJ impact ...... 187 Figure 7-13: Absorbed energy by waterbomb under 20 mJ impact ...... 188 Figure 7-14: Absorbed energy by waterbomb upside down under 20 mJ impact ...... 188 Figure 7-15: Absorbed energy by regular honeycomb under 20 mJ impact ...... 189 Figure 7-16: Absorbed energy by chevron under 20 mJ impact ...... 189 Figure 7-17: Absorbed energy by inverted honeycomb under 30 mJ impact ...... 191 Figure 7-18: Absorbed energy by waterbomb under 30 mJ impact ...... 191 Figure 7-19: Absorbed energy by waterbomb upside down under 30 mJ impact ...... 192 Figure 7-20: Absorbed energy by regular honeycomb under 30 mJ impact ...... 192 Figure 7-21: Absorbed energy by chevron under 30 mJ impact ...... 193 Figure 7-22: Absorbed energy by inverted honeycomb under 40 mJ impact ...... 194 Figure 7-23: Absorbed energy by waterbomb under 40 mJ impact ...... 194 Figure 7-24: Absorbed energy by waterbomb upside down under 40 mJ impact ...... 195 Figure 7-25: Absorbed energy by regular honeycomb under 40 mJ impact ...... 196 Figure 7-26: Absorbed energy by chevron under 40 mJ impact ...... 196 Figure 7-27: Absorbed energy by inverted honeycomb under 50 mJ impact ...... 197 Figure 7-28: Absorbed energy by waterbomb under 50 mJ impact ...... 198 Figure 7-29: Absorbed energy by waterbomb upside down under 50 mJ impact ...... 198 Figure 7-30: Absorbed energy by regular honeycomb under 50 mJ impact ...... 199 Figure 7-31: Absorbed energy by chevron under 50 mJ impact ...... 200 Figure 7-32: Absorbed energy of inverted honeycomb under 10 mJ to 50 mJ impact ...... 201 Figure 7-33: Absorbed energy of waterbomb under 10 mJ to 50 mJ impact...... 202 Figure 7-34: Absorbed energy of waterbomb upside down under 10 mJ to 50 mJ impact ...... 202 Figure 7-35: Absorbed energy of regular honeycomb under 10 mJ to 50 mJ impact ...... 203 Figure 7-36: Absorbed energy of chevron under 10 mJ to 50 mJ impact ...... 203 Figure 7-37: Absorbed energy of all models under 10 mJ to 50 mJ impact ...... 204 Figure 7-38: Average of absorbed energy by all models ...... 206

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Chapter 1

1. Introduction

Sandwich panels are used in several industries. A sandwich structure is made of a light weight, lower strength core sandwiched between two thin, stiff faces. The combination of light weight and desired mechanical properties makes these structures attractive for construction, automotive, aerospace and many other industries [1].

Conventional honeycombs (Figure 1-1), which have high stiffness and are lightweight are widely used as the cores of sandwich structures.

One problem with regular honeycomb is that when humidity accumulates in the cells, it cannot escape and the result is that the structure becomes heavy. Foldcore geometries offer ventilation channels to overcome this problem (Figure 1-2). Also, other advantages, such as the simpler and faster manufacturing process compared with honeycombs, have motivated researchers to study this sort of cores. One of the materials that are used for manufacturing foldcores is aramid papers which have random fiber distribution are saturated with phenolic resin, and are manufactured through a continuous process. Kevlar® and Nomex® are two types of aramid papers and are used in the production of foldcores and conventional honeycomb cores [2].

Figure 1-1: Honeycomb core structure [2]

Figure 1-2: Chevron foldcore structure [3]

Figure 1-3: Modified Chevron foldcore structure [3]

Origami is the Japanese art of folding paper into decorative shapes and figures.

The most common geometry in foldcore research – the so-called – chevron, Figure 1-2, pattern is well-known in origami and is known as Miura. A modified version of this geometry, referred to in the foldcore literature as the modified chevron, Figure 1-3, has also been studied [4]. In this project, we will explore the feasibility of other origami- inspired foldcore geometries. Specifically, we will focus on patterns that exhibit negative

Poisson’s ratios. Negative Poisson’s ratio materials (i.e. auxetic materials have many attractive structural properties). We will investigate whether folded auxetic cores can be used to achieve structural sandwich panels with enhanced properties.

Figure 1-4: Flat-folded (right) and curved (left) Waterbomb [3]

One problem with honeycomb and chevron cores is that they cannot be easily formed into a dome-shaped (i.e. synclastic) structure. Figure 1-4 shows waterbomb tessellation, with negative a Poisson’s ratio. This geometry has been inspired by the origami methods and is very similar to the inverted honeycomb, however the inverted honeycomb is an extruded form of the two-dimensional reentrant honeycomb pattern and waterbomb is a three-dimensional structure. Waterbomb, when used as the core of a sandwich panel, unlike conventional honeycombs and chevrons, might easily form a dome. Patterns like waterbomb might assist manufacturing of curved cores of sandwich panels in a continuous process. Therefore there would be no need to rivet or weld flat sandwich panels for the curved structures such as dome shape roofs, helmets or airplane cones.

The ultimate goal of this study is to investigate the response of these structures to compression, shear, bending and impact loading and to compare the mechanical properties with other core geometries. Then, researchers and manufacturers would be able to pick the best-performing structure under each load for further studies.

Chapter 2

2. Literature Review

2.1. Auxetic Materials

Auxetic materials are those that possess negative Poisson’s ratios. Poisson’s ratio

(ν) is one of the elastic properties of materials. The Poisson’s ratio is expressed as:

휖푖 휈푖푗 = − 휖푗 where 휖푖 is transverse strain,

and 휖푗 is longitudinal strain.

Most of materials expand in the lateral direction under compression loadings, and contract in the lateral direction when being stretched. This behavior is a consequence of having a positive Poisson’s ratio which most materials, like metals, do possess.

2.1.1. Causes of auxeticity

One leading cause of negative Poisson’s ratio is the construction of the material itself. In other words, microstructure is a main reason of auxeticity. In 1987, Lakes was the first to observe negative Poisson’s ratio behavior in polyurethane under longitudinal deformation [5]. Polyurethane foams were the first man-made materials with a negative

Poisson’s ratio. This was done by creation of reentrant cellular structures. Figure 2-1 shows the ideal cellular structure of a reentrant honeycomb.

Figure 2-1: Ideal cellular structure of reentrant honeycombs [2]

An example of an auxetic microporous polymer is PTFE [6]. In the microstructure network of this polymer, the negative Poisson’s ratio is an outcome of the molecular bonding deformation under an applied load. Other examples of auxetic polymers are

UHMWPE [7]–[9] and PP [10] which by going through sintering and extrusion processing, achieve negative Poisson’s ratios. While isotropic materials possess a

Poison’s ratio between -1 to 0.5, an unusual Poisson’s ratio of -12 has even been reported for these materials [6].

Some laminated reinforced composites have also been reported to have negative

Poisson’s ratios. [11]. Also, angle-ply composites made of unidirectional glass fiber

layers in epoxy resin matrices are proved to have large negative out of plane Poisson’s ratios [12].

For molecular materials, auxeticity can be attained by redeveloping the construction on the scale of molecules. Figure 2-2 schematically shows an example of auxeticity caused by microstructures. Zeolites and silicon dioxide are two examples of such materials.

Figure 2-2: Schematic of inverted honeycomb showing transverse expansion when stretched (top) and (1, 4)-refexyne with a negative Poisson’s ratio (bottom) [2].

2.1.2. Benefits

In theory auxetic materials have enhanced mechanical properties. For example, isotropic materials’ hardness is inversely proportional to the magnitude of (1 − ν2).

Therefore, by classical elasticity theory, the greatest hardness of a positive Poisson’s ratio material is reached by having a Poisson’s ratio value of +1/2. Auxetic materials with

similar Young’s modulus and a Poisson’s ratio of less than -0.5 would have even greater hardness.

Auxeticity has considerable effects on materials’ mechanical properties, and some remarkable behaviors can appear in the occurrence of negative Poisson’s ratio.

Enhancement of the properties like in plane strain resistance in the case of fracture and shear modulus [13] fracture resistance [5], [14] and acoustic response [15] are expected by the classical theory of elasticity due to the auxeticity.

Materials with negative Poisson’s ratios also have increased elastic shear modulus and the dynamic loss tangent when compared with positive Poisson’s ratio material with the same Young’s modulus [16].

Figure 2-3: Material’s density below the impact point is different in auxetic and non- auxetic materials [2].

Auxetic materials also have enhanced indentation resistance also. In comparison with the conventional honeycombs with the equal density, the yield strength of reentrant copper foams is higher while the stiffness is lower [17]. The effects of the Poisson’s ratio’s sign on indentation resistance can be seen in Figure 2-3. In conventional materials,

the density decreases immediately under the impact point; however, in the case of auxetic materials, lateral contraction leads the material movement towards the impact area and accordingly, results in an increase of hardness [2].

Furthermore, materials’ viscoelastic performances improve in the existence of auxeticity. A numerical study by Scarpa et al. proved that the moduli of two-phase composite foam increased when reentrant honeycomb structures were employed. Also, this study presented a possibility to improve both stiffness and modal loss factors by using auxetic honeycombs [18].

Additionally, deformation behaviors, which are the most significant aspect of the auxetic materials’ performance, obviously, change with the sign of the Poisson’s ratio. A as Poisson’s ratio of -0.8 of reentrant copper foam increase the inhomogeneous micro- deformation in comparison with the regular shape of the foam [19]. All other behaviors of auxetic materials are consequences of deformation behaviors.

Lastly, some other material properties have been shown to be more appropriate when the material is auxetic. A brief list of those includes capacity of damping

[20],absorption behaviors [21], [22] and the transverse shear moduli [16].

In the case of honeycombs, an increase of angles between cell ribs will result in reduction of the structure’s Young’s modulus, and it was learned that the regular honeycombs provide a smaller value of transverse Young’s modulus when compared with the inverted honeycombs [16]. Inverted honeycomb configuration can be achieved by applying some modifications to the structure of regular honeycomb cells. Several studies have been done focusing on numerous features of inverted honeycombs. For

example, material constants [23], geometric effects [24], density variations [25], electromagnetic properties and mechanical properties [16] have been investigated.

2.2. Sandwich panels

One kind of materials which often show auxeticity are composites. Different causes can result in negative effective Poisson’s ratios for composite materials. The main cause is the auxeticity of the reinforcing ingredients itself. However, auxeticity, also, can be accomplished through using a positive Poisson’s ratio material as the core of sandwich structures. This happens when geometries which their effective Poisson’s ratios are negative are employed [2]. As mentioned before, inverted honeycombs are one of the geometries for showing auxeticity when used as the core of a sandwich panel. Some have also shown interest in studying chevron patterns because of their capability of humidity disposal and continuing to remain a very low ratio of weight to strength.

Sandwich panels, as stated in the introduction, are widely used in many industries.

For example, in the aerospace industry, a very close attention in making the shuttle light while keeping the strength high enough is strictly required which can be delivered by using the sandwich panels with low weight to strength ratios. This, also, applies to other transportation industries like automotive industry. Sandwich structures is one of the priorities of the automobile designers.

2.2.1. Benefits of Auxetic Cores

Numerous investigations have been done on the different performances of sandwich panels with different core shapes and materials. Through the thickness

Poisson’s ratio, buckling strength, compression responses, shear stiffness and impact are

some of the investigated properties. Aramid papers, carbon fiber reinforced plastics

(CFRP) and glass fiber reinforced plastics (GFRP), have been studied due to their desirable behaviors as the cores of the sandwich structures. The core geometry is another field of study in the case the sandwich panels [2].

The through-the-thickness Poisson’s ratio of angle-ply laminates is different from that of off-axis lamina. This happens to the in plane Poisson’s ratios of laminates when compared with laminas [26]. Since the Poisson’s ratio is “an even function of θ for unidirectional off-axis laminae”, 휈푥푧 is positive for both positive and negative values of the angle of fiber orientations [27]. However, a certain combination of positive and negative theta layers results in a negative Poisson’s ratio. This indicates that all relationships between constraining layers must be considered [27].

Under through the thickness loading, in the case of buckling, conventional honeycombs act as sets of plates. The resistance of honeycombs against buckling is not significantly larger than the total resistance of the same amount of independent plates

[28]. When flexure out of plane loading is applied, the regular honeycombs deform into anticlastic curvatures. Auxeticity nature allows the reentrant honeycombs to form synclastic curvatures that is anticipated in forming dome shape structures. Also, when auxetic honeycombs are used in these applications the risk of local damage caused by making conventional honeycombs form like a dome considerably decrease [28], [29].

Miller et al. showed that the inverted honeycombs’ stress peak can tolerate almost twice than which of the hexagonal honeycombs with the same ribs lengths [28]. The transverse expansion of the ribs under through the thickness loading is, obviously, caused by positive Poisson’s ratios. If the compression is in both planar axes, the number of

available deformation modes before buckling starts decreases, and as a result the peak load at buckling decreases as well. In summary, in auxetic geometries the ribs experience contraction; thus, inverted honeycombs possess larger buckling resistance than the as dense regular honeycombs [28].

In the case of impact, Buitrago et al. [30] numerically investigated the damage of composite sandwich structures exposed to high-velocity. They also performed experiments to verify the simulation. The difference between simulation results and experimental results were reported to be less than 2% which validated the numerical results. The tests were done under a comprehensive series of impact velocities and the abilities of the facesheets and cores to absorb energy was studied. They demonstrated that the facesheets absorbed a much greater extent of impact energy when compared with the core. To be more detailed, they showed that about 85% of the impact energy, for velocities more than 250 m/s, was absorbed by the facesheets. The front facesheet absorbed slightly more energy than the back facesheet. When the velocity became closer to the ballistic limit, the core absorbed 10%-20% of the energy whereas the impact energy absorbed by the front facesheet was about 60%. They also showed that the mechanisms of the energy-absorption is different in the facesheets and the core. While a small area plastic deformation was observed in the core, impact resulted in fiber breakage in the facesheets [30].

In another study on impact, Zhou et al. [31] performed an experimental and numerical study on response of sandwich panels with graded foam cores caused by the low velocity impact. It was shown that shear in the core was the common reason of the failure of the sandwich panel. Finite element simulations exactly predicted the failure

mode. It also accurately anticipated the perforation energy and load–displacement associated with the impact. They proved that locating denser foam cores alongside the top panel would enhance perforation resistance comparative to sandwich structures with the denser foam cores touching the bottom panel. They, also, observed the advantages of placing the less dense foam in the center of the core and a ductile foam against the lower facesheet [31]

Caprino et al. [32] performed low velocity impact tests to investigate damage in sandwich structures. They used an instrumented ball drop machine to drop balls on the sandwich panels with glass/ polyester facesheets and polyvinylchloride (PVC) foam core.

Tests were done on different cores with different materials and densities. One important output of the experiment was that the extent of damage is actually irrelevant to the thickness and material of the core and is a direct result of perforation energy up to the perforation threshold. Once the energy passes this point the extent of damage stays the same.

Some research also been done to improve the impact properties of carbon fiber sandwich composites which are of the most desirable composites because of having high stiffness and low weight but low impact resistance. Gustin et al. [33] examined three different materials to add to the face panels of those sandwich structures in order to improve their impact performance. They tried adding carbon fiber, Kevlar or a combination of both to the facesheets and reported gaining enhanced impact resistances by adding each of the mentioned materials to the panels. Further improvements can be found in [33].

Whitty et al. numerically studied the built-up stress, caused by thermal loading conditions and hydrostatic stress, in the honeycomb cores of sandwich panel composites, and confirmed that a change from conventional honeycombs to auxetic honeycombs minimizes the stress build-up in the ribs which is induced thermally [34].

As stated earlier, foldcores have recently become more interesting for researchers as they offer many desired features. In the following paragraphs some comparisons between different foldcores are mentioned. All of the introduced models are inspired by Origami.

Schenk et al. [35] revealed some facts about two origami models, Miura (chevron) and Eggbox. These models are shells which exhibit several noteworthy properties. They display benefits of using folded sheets of paper when the shape and Gaussian curvature were changed but the material, itself, was not stretched [35]. These models, then, could be very useful in the architecture, where they can be used as covers for doubly-curved shells. The geometries can have either positive or negative Poisson’s ratios under tensile and bending loads. The crease patterns have the capability to deform simply in some modes while maintaining a reasonable stiff in others. Supplying flexibility and rigidity in different modes is much favorable in morphing structures. In fact, the models can exhibit such properties which are able to open and close the folds. Folded structures, characteristically, show higher stiffness while maintaining the weight low. As a result, origami patterns are very beneficial when used in manufacturing of sandwich structures, wrapping solar sails [36], medical stents [37], emergency shelters [38], shock absorbing devices [39], and packaging materials [40].

By studying the response of the models under different loads, some observations could be made. To begin with, both models bend to form a saddle-shaped configuration

and change their global Gaussian curvatures. Next, Eggbox demonstrate a positive

Poisson’s ratio whereas Chevron displays auxeticity in extension. Furthermore, in the case of bending, Eggbox forms a synclastic (cylindrical or spherical) shape, while

Chevron deforms into anticlastic (saddle-shaped) configuration. The models contradictory actions point toward that the geometry has an superior level of significance in the performance of the models when compared with the material’s property [35].

The most common types of sandwich panels consist of thin and strong sheets of materials with high stiffness and density. A thick layer of a light material with lesser stiffness separates the facesheets. This layer is called core. To be efficient, the core’s weight should be almost equal to the weight of the skins [41]. Evidently, sandwich structures provide a much higher bending stiffness than the same weight plates made of the same material [41].

The core plays a very significant role in the performance of the sandwich panels.

It maintains the facesheets at the same distance at each and every point. It must also have a reasonable stiffness to prevent the panels from sliding while being under bending and shear loading conditions. In addition, the core functions as the mean to keep the facesheets flat in order to avoid local buckling [41].

Cores are made of several different materials with some different designs and geometries. Some of the most common types of the cores are presented and briefly introduced in the following paragraphs.

Honeycombs, Figure 2-4, are one of the most conventional geometries used as the cores for sandwich panels. This type of core is extensively used for applications, which require a high strength in compression and buckling. For example sandwich panels with

honeycomb cores are widely employed in manufacturing of airplanes [41] and constructing bridges [42]. One of the main drawbacks of honeycombs is that manufacturing of honeycomb cores is a complex and expensive process. Also, when they accumulate humidity it cannot escape and the core becomes very heavy [43].

Figure 2-4: Schematic of conventional honeycomb cores

Corrugated cores, Figure 2-5, are another type of core commonly used in industry.

The corrugated core is typically a grooved sheet of metal attached to the top and bottom facesheets. One application of corrugated cores is that they can be an alternative to honeycombs for shipbuilding [44], [45], [46]. Corrugated core sandwich panels perform well under impulsive and blast loadings [47]. One of the main concerns in application of this type of cores in morphing structures is that the facesheets do not bend smoothly [48].

Figure 2-5: Schematic of corrugated cores

Foams, Figure 2-6, are also used as the core in the sandwich structures. These cores have applications in many industries including aerospace [49] and construction

[50]. Foam cores are widely known for their resistance against penetration. Therefore, many of aircraft manufacturers use sandwich panels with foam cores in manufacturing the nose of planes to prevent the perforation caused by the bird strikes [49]. Besides being extremely light, foam cores are perfect to be used in walls and roofs in the buildings, since they display a great capability in thermal insulation [50]. However, one disadvantage is that when there is a local damage between the core and the panels the ability of carrying the loads considerably decreases [51].

Figure 2-6: Schematic of foam cores

Woven fabric cores, Figure 2-7, are another type of cores, which have various applications. In railroad and automotive industries, sandwich panels with woven cores are employed in building the train floors as well as the top of the convertible super sport cars and side walls of trucks [52]. Sandwich panels having woven fabric cores take advantage of a very high debonding resistance between the facesheets and the core [53]. Also, they offer a reasonably high energy absorption ability [54].

Figure 2-7: Schematic of woven cores

Foldcores, Figure 2-8, are relatively new into sandwich structure industry. This kind of cores are produced in a continuous process of folding sheets of materials that is inspired by the art of Origami. They are currently being studied and examined numerically and experimentally, and have already been used for their applications in aerospace and automotive industries, since they offer the desired performances while the weight of the core, and consequently the whole structure, reduces reasonably [55].

Figure 2-8: Schematic of a foldcore

2.2.2. Materials

Heimbs [56] experimentally and numerically compared the CFRP and Kevlar® aramid paper performances under flatwise compression. The materials were assumed to be isotropic. The experiments validated the modeling results for cell wall deformations and stress-strain curves. Similarly, WT-plane curves match in the experimental results and simulations with a good precision whereas the TL-plane is inconsistent. Since the model is validated by experiments, the results of simulation is valuable for the other

loading conditions like “compressive, tensile and shear behavior in all in-plane and out- of-plane material directions using virtual tests” [56].

2.3. Modeling

To run the simulations, one must first know the mechanical properties of the used material. Finding the compressive properties of a lone sheet of paper is very difficult.

There are some standard methods for this. One is the ring-crush method, which Heimbs et al. [4] used to determine aramid paper properties, and ran the simulations based on them.

Thus, the obtained mechanical behaviors under tensile and compressive loading are based on the valuable given properties to the software.

In many cases, gaining data through experimental tests is not as easy as obtaining data through simulations. In the case of Kevlar® chevrons and CFRP foldcores, models studied under tensile, compressive and shear loading conditions recognized the stress- strain performance in all materials [56].

As Heimbs [56] indicated, these studies had two substantial outcomes. First of all, virtual simulations results on foldcore geometries are considerably reliable. Secondly, the use of the foldcore as the cores for the sandwich panels is practical; In conclusion, folding materials is a reasonable method and using other foldcore geometries might be beneficial as well.

Finding a method for modeling patterns to be ready for be run in the simulations is the first step to use origami models in engineering. To start with, the origami patterns should be generated in the proper software; this simplifies the understanding of the structures and geometries very much. An approach suggested by Miyazaki et al. [57] includes a series of steps to fold sheets of paper in order to finish up with the expected 20

geometry. Oripa [58] is one drawing software that is appropriate for drawing and exporting crease patterns to other softwares. However, Oripa does not let the designer control the middle steps of folding, and only gives the final shape of the folded model.

The simulation method designed by Tachi [59] is based on rigid origami. In this system, the model forms according to folding lines angles and the restraint space of folding. Therefore, the locations of the virtual cuttings do not affect the overall motion.

Also, the adjustment of angle and crease line option is provided in this. A “closed vertex” is the point that three adjacent crease lines meet, while the center line is the symmetrical line for the outer two and their mountain-valley assignments are contrary [59]. One problem is that the motion of fold lines is locked by the new fold line in closed vertices.

Tachi proposed two solutions for avoiding closed vertices. By either adding a new line passing the vertex, or changing the inside angle in a way that the symmetry disappears one can overcome this problem. Hence, enough degrees of freedom can be achieved and the singularity problem can be solved. The total degree of freedom is a function of N and

M which are the number of edges and vertices, respectively. While counting the edges and vertices, only those inside the drawing, not on the perimeter, are counted. The total

DOF (degree of freedom) of the model is equal to N-3M. Triangulation is a method of escaping the absence of enough DOF which stops the model from folding rigidly. When singularity is the problem, the vertices are relocated in a way that the inside angles decrease.

Kirigami is a combination of ply-cut process and Origami. This method has newly become of interest in manufacturing of the foldcores with complex geometries with single prepreg sheets of composite papers like aramid paper sheets [60]. This technique

involves mixing cuts with valley/mountain folds to generate the foldcore. To be more comprehensive, the paper is cut into the slits with the known width and slits are folded into valley and mountains [61]. Hou et al. studied the compressive modulus and strength in the cores made with Kirigami method and showed that the compressive modulus and strength of the stabilized honeycombs are greater than which of the plain honeycombs.

They also indicated that graded sandwich structures under quasi-static edge-wise loading present enhanced mechanical properties such as strength and modulus.

Chapter 3

3. Motivation and Objectives, and Procedure

3.1. Motivation and objectives

Interest in investigating the practicality and potential benefits of employing the inverted honeycombs, waterbomb chevron and regular honeycomb models as the core for the sandwich panels was the main motivation for this study. These geometries exhibit negative Poisson’s ratios under several loading conditions, which could be beneficial in many applications.

The objective of study is to find the effective Young’s modulus and effective

Poisson’s ratio of the abovementioned geometries under compressive loading conditions in the z direction, in the x direction and in the y direction, the effective shear modulus, and to investigate the performance of origami inspired cores in impact, and to compare the effective properties of the different geometries. Also, further, the curvature of these models under two different bending loads and also the energy absorption of the models under five different impact loads will be explored.

The effective Young’s modulus is found based on the following equation:

∑ 퐹 퐴 퐸푒푓푓 = ℰ푙 where ∑F is the sum of the reaction forces at the supports; A is the effective area of the model perpendicular to the force direction and ℰ푙 is strain along the force direction.

The effective Poisson’s ratio was found based on the following relation:

ν=− ℰ푡 ℰ푙 where ℰ푡 is the transverse strain and ℰ푙 the longitudinal strain.

3.2. Procedure

In many cases, gaining data through experimental tests is not as easy as obtaining data through simulations. As Heimbs [56] indicated, these studies had two substantial outcomes. First of all, virtual simulations results on foldcore geometries are considerably reliable. Secondly, the use of the foldcore as the cores for the sandwich panels is practical; In conclusion, folding materials is a reasonable method and using other foldcore geometries might be beneficial as well.

A simulation approach which has shown to be reliable should be designed and followed. The approach employed for this part of the study is very similar to the one that

Heimbs [56] designed for compressive and shear loads in his work. He experimentally and numerically compared the CFRP and Kevlar® aramid paper performances under flatwise compression. The materials were assumed to be isotropic. The experiments validated the modeling results for deformations and stress-strain curves. Similarly, WT- plane curves match in the experimental results and simulations with a good precision.

Since the model is validated by experiments, the results of simulation are valuable for the

other loading conditions like “compressive, tensile and shear behavior in all in-plane and out-of-plane material directions using virtual tests” [56].

In this approach the development of the models for simulations of cellular core structures contains assigning of material properties, generating the mesh, and applying the appropriate boundary and loading conditions.

Waterbomb and chevron were geometrically created in Rhinoceros drawing software and imported to Comsol Multiphysics. Regular honeycomb and inverted honeycomb were built and finalized in Comsol Multiphysics. Two sandwich facesheets are generated on the top and bottom side of the core, where the loading and boundary conditions are applied. The entire model is meshed with extra fine triangular shell elements based on the defined element size. The core and the facesheets are coupled by a contact formulation. Then the boundary conditions (described later) are applied to the models. In this approach quarter-symmetry is used to reduce computational cost.

Therefore, the essential boundary conditions are included at the symmetry surfaces.

On the top and bottom surface of the core structure a thin skin panel was generated, where the boundary and loading conditions were applied. The assembly and finalizing of the core structure and the facesheets was achieved by Form A Union contact feature in Comsol Multiphysics. The boundary and loading conditions are explained in details for each model further in this section. To stick with the experimentally verified approach in [56], a displacement function as the loading condition was defined. The direction of this displacement was chosen with respect to the anticipated compression, or shear loading situation. The purpose for the employment of these extra plates as an alternative of assigning the boundary conditions directly to the upper and lower nodes of

the core model was to have a contact surface to prevent the cell walls from impractical out-of-plane deformations during the loading procedure.

3.2.1. Material

Although all of the results are normalized and therefore, the kind of used material did not affect the effective properties, to run the simulation we had to provide the basic properties of a given material to the simulation program. Thus, this work is done based on the properties of Aramet® papers manufactured by Aramid, Ltd [62]. Young’s modulus, density and Poisson’s ratio are the needed mechanical properties for the linear elastic analysis. Density is provided by the manufacturer and Poisson’s ratio of 0.3 is a reasonable value for Aramid. The compressive modulus of each paper used in this work is based on the experiment done in [2] which has followed the “Ring crush of paperboard“ method, TAPPI 822 [63]. A summary of the material’s mechanical properties and thickness can be seen in Table 3.1.

Table 3-1: Material’s properties

Name Thickness Average Young’s Density ( 푘푔 ) Poisson’s 푚3 (mm) Modulus (MPa) Ratio (Supplied by (From [2]) Manufacturer)

Aramet® 7 mil 0.18 185 960 0.3

3.2.2. Geometry

Models have four geometries, inverted honeycomb, waterbomb, chevron and regular honeycomb. Waterbomb and chevron are generated in Rhinoceros 5.0 [64] and

are saved as the *.stl format and finally imported to Comsol Multiphysics 5.1 [65].

Inverted honeycomb and regular are drawn in workplane in Comsol and are extruded to the desired height. The models then are joined and formed a union.

The models are studied under axial compression in the x, y and z directions separately as well as shear in the x and y directions separately. Only one load in one direction is applied at the time and the shell physics is employed with a stationary loading condition. The best mesh suitable for the shell physics is triangular and the size is chosen to be extra fine. The models are initially studied with extremely fine mesh case but the data is only 0.1% more precise than extra fine mesh case and but it is about 100% more time consuming. Therefore, using extra fine mesh case was reasonable. The coordinate used for this study is Cartesian while the predefined length unit was selected to be meter.

In the following sections, the loading and boundary conditions for each geometry and loading conditions are introduced.

Figure 3-4: Waterbomb generated in Rhinoceros [64]

Figure 3-5: Imported Waterbomb to Comsol

Figure 3-6: Inverted honeycomb generated in Comsol 28

Figure 3-7: Imported Chevron to Comsol

Figure 3-8: Regular honeycomb generated in Comsol

3.2.3. Loading

3.2.3.1. Compression in the z direction

In the case of compression in the z direction, the boundary loading conditions employed for all models are very similar. The boundary conditions are described in detail for each geometry below.

3.2.3.1.1. Inverted honeycomb

The model dimensions are:

퐿푥= 0.40 m

퐿푦= 0.40 m

퐿푧= 0.02 m

Three different displacements which resulted in 3 desired strains are separately applied to each model with a defined thickness of .18 millimeters.

In the case of compression in the z direction, to apply the boundary conditions, the surfaces on the plane of y=0 are allowed to move only in the x direction. The edges on the plane of x=0 are allowed to move only along the y direction. The edges located on the plane of z=0 are allowed to move along the x and y directions, but not along the z direction. Known displacement of -0.001 m is applied to the edges on z=퐿푧 , which results in a 5% strain along the z dimension.

For sandwich panels, the edges and/or surface on the planes of x=0, y=0 and z=0 are allowed to move on the same plane only.

Figure 3-9: Inverted honeycomb in Comsol Multiphysics [65]

3.2.3.1.2. Waterbomb

Next studied geometry was the waterbomb. The initial dimensions of the models are:

퐿푥= 0.40 m

퐿푦= 0.40 m

퐿푧= 0.02 cm

The boundary and loading conditions for this geometry are quite similar to those of the previous model. The edges on the plane of y=0 are constrained to move only in the x direction, and the edges on the plane of x=0 are allowed to move along the y direction.

The points on the plane of z=0 were acceptable to move on the plane of xy. The

displacements applied to the edges on the z=퐿푧, producing strain of 5% is equal to -0.001 m.

For sandwich panels, the edges and/or surface on the planes of x=0, y=0 and z=0 are allowed to move on the same plane only.

Figure 3-10: Waterbomb in Comsol Multiphysics [65]

3.2.3.1.3. Regular honeycomb

The model dimensions are:

퐿푥= 0.80 m

퐿푦= 0.40 m

퐿푧= 0.02 m

Three different displacements which resulted in 3 desired strains are separately applied to each model with a defined thickness of .18 millimeters.

For sandwich panels, the edges and/or surface on the planes of x=0, y=0 and z=0 are allowed to move on the same plane only.

Figure 3-10: Regular honeycomb in Comsol Multiphysics [65]

3.2.3.1.4. Chevron

The model dimensions are:

퐿푥= 0.40 m

퐿푦= 0.221 m

퐿푧= 0.028 m

Three different displacements which resulted in 3 desired strains are separately applied to each model with a defined thickness of .18 millimeters.

In the case of compression in the z direction, to apply the boundary conditions, the points on the plane of y=0 are allowed to move only in the x direction. The edges on the plane of x=0 are allowed to move only along the y direction. The edges located on the plane of z=0 are allowed to move along the x and y directions, but not along the z direction.

Known displacement of -0.0014 m is applied to the edges on z=퐿푧 , which results in a 5% strain along the z dimension.

For sandwich panels, the edges and/or surface on the planes of x=0, y=0 and z=0 are allowed to move on the same plane only. Also the strain is 1%.

Figure 3-12: Chevron in Comsol Multiphysics [65]

3.2.3.2. Compression in the x direction

In the case of compression in the x direction, all models’ dimensions and boundary conditions are the same as compression in the z direction. However, the loading conditions must be changed. These conditions will be discussed in detail in the following sections.

3.2.3.2.1. Inverted Honeycomb

The displacement of -0.02 m is applied to the edges on the plane of x=퐿푥 to produce 5% strain in the x direction.

3.2.3.2.2 Waterbomb

A displacement of -0.02 m is applied to the edges on the plane of x=퐿푥 are to result in strain of 5% in the x direction.

3.2.3.2.3 Regular honeycomb

A displacement of -0.04 m is applied to the edges on the plane of x=퐿푥 are to result in strain of 5% in the x direction.

3.2.3.2.2 Chevron

A displacement of -0.02 m is applied to the edges on the plane of x=퐿푥 are to result in strain of 5% in the x direction.

3.2.3.3. Compression in the y direction

In the case of compression in the y direction, for a third time, the models remained their initial dimensions and boundary conditions, and loading conditions are changed in the ways discussed in the following sections.

3.2.3.3.1. Inverted honeycomb

A displacement producing 5% is equal to -0.02 m and is applied to the edges at y=퐿푦 .

3.2.3.3.2. Waterbomb

A displacement equal to -0.02 m is applied to the edges at the plane of y=퐿푦 to create a 5% strain in the y direction.

3.2.3.3.3. Regular honeycomb

A displacement producing 5% is equal to -0.02 m and is applied to the edges at y=퐿푦 .

3.2.3.3.4. Chevron

A displacement causing 5% is equal to -0.01105 m and is applied to the points at y=퐿푦 .

3.2.3.4. Shear in the x direction

In the case of shear in the x direction, for inverted honeycomb, regular honeycomb and chevron all the edges on the plane of xy are fixed, and for the waterbomb all the points and edges on the plane of z=0 are fixed and the loads are applied to the edges at the plane of z=퐿푧.

For sandwich panels, the surface on the plane of z=0 is fixed and the strain is 1%..

3.2.3.4.1 Inverted honeycomb

A displacement of -0.001 m is applied to the edges at the plane of z=퐿푧 to result in shear strain of 5%.

For sandwich panels, the surface on the plane of z=0 is fixed and the strain is 1%.

3.2.3.4.2 Waterbomb

A displacements of -0.001 m is applied to the edges at the plane of z=퐿푧 to end up with shear strain of 5%.

For sandwich panels, the surface on the plane of z=0 is fixed and the strain is 1%..

3.2.3.4.3 Regular honeycomb

A displacements of -0.001 m is applied to the edges at the plane of z=퐿푧 to result with shear strain of 5%.

For sandwich panels, the surface on the plane of z=0 is fixed and the strain is 1%.

3.2.3.4.4 Chevron

A displacements of -0.0014 m is applied to the edges at the plane of z=퐿푧 to end up with shear strain of 5%.

For sandwich panels, the surface on the plane of z=0 is fixed.

3.2.3.5. Shear in the y direction

In the case of shear in the y direction, similar to shear in the x direction, all edges on the plane of z=0 are fixed and the loads are applied to the edges at the plane of z=퐿푧.

For sandwich panels, the surface on the plane of z=0 is fixed.

3.2.3.5.1. Inverted honeycomb

Shear strains of 5% is created by displacement of -0.001 m, which is applied to the edges at the plane of z=퐿푧. For sandwich panels the strain is 1%.

3.2.3.5.2. Waterbomb

Shear strain of 5% resulted in displacement of -0.001, is applied to the points at the plane of z=퐿푧. For sandwich panels the strain is 1%.

3.2.3.5.3 Regular honeycomb

A displacements of -0.001 m is applied to the edges at the plane of z=퐿푧 to result with shear strain of 5%. For sandwich panels the strain is 1%.

3.2.3.5.4 Chevron

A displacements of -0.0014 m is applied to the edges at the plane of z=퐿푧 to end up with shear strain of 5%. For sandwich panels the strain is 1%.

Chapter 4

4. Compression and Shear for the Cores

In this chapter, the test results of all cores will be presented and analyzed. The results are presented in a way that all models’ data of compression tests come first followed by the data of shear tests. All simulations are done using the material properties for the Aramet® papers called 7 mil (0.18 mm thick). For all models results for 7 mil papers under a 5% strain are presented in the section 4.1 and the necessary data for analysis is given in the section 4.2.

4.1. Results

4.1.1. Compression in the z direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were also described in section 3.2.3.

4.1.1.1. Inverted honeycomb

Based on the test conditions discussed in section 3.2.3.1.1, Figure 4-1 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% compressive strain in the z direction.

Figure 4-1: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the z direction.

4.1.1.2 Waterbomb

The test conditions for Waterbomb were introduced in section 3.2.3.1.2. Figure 4-

2 shows the deformation and stress distribution for Waterbomb 7 mil paper under 5% compressive strain in the z direction.

Figure 4-2: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the z direction.

4.1.1.3. Regular honeycomb

Based on the test conditions discussed in section 3.2.3.1.3, Figure 4-3 shows the deformation and stress distribution for regular honeycomb 7 mil paper under 5% compressive strain in the z direction.

Figure 4-3: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the z direction.

4.1.1.4. Chevron

The test conditions for Waterbomb were introduced in section 3.2.3.1.4. Figure 4-

4 shows the deformation and stress distribution for Waterbomb 7 mil paper under 5% compressive strain in the z direction.

Figure 4-4: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Chevron model under 5% compressive strain in the z direction.

4.1.2. Compression in the x direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were, also, introduced in section 3.2.3.

4.1.2.1. Inverted honeycomb

Based on the test conditions discussed in section 3.2.3.2.1, Figure 4-3 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% compressive strain in the x direction.

Figure 4-5: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the x direction.

4.1.2.2. Waterbomb

Based on the test conditions discussed in section 3.2.3.2.2, Figure 4-4 shows the deformation and stress distribution for Waterbomb 7 mil paper under 5% compressive strain in the x direction.

Figure 4-6: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the x direction.

4.1.2.3. Regular honeycomb

Based on the test conditions discussed in section 3.2.3.2.3, Figure 4-7 shows the deformation and stress distribution for regular honeycomb 7 mil paper under 5% compressive strain in the x direction.

Figure 4-7: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the x direction.

4.1.2.4. Chevron

Based on the test conditions discussed in section 3.2.3.2.4, Figure 4-8 shows the deformation and stress distribution for chevron 7 mil paper under 5% compressive strain in the x direction.

Figure 4-8: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% compressive strain in the x direction.

4.1.3. Compression in the y direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were also introduced in section 3.2.3.

4.1.3.1. Inverted honeycomb

Based on the test conditions discussed in section 3.2.3.3.1, Figure 4-9 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% compressive strain in the y direction.

Figure 4-9: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% compressive strain in the y direction.

4.1.3.2. Waterbomb

Based on the test conditions discussed in section 3.2.3.3.3, Figure 4-10 shows the deformation and stress distribution for Waterbomb 7 mil paper under 5% compressive strain in the y direction.

Figure 4-10: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% compressive strain in the y direction.

4.1.3.3. Regular honeycomb

Based on the test conditions discussed in section 3.2.3.3.3, Figure 4-11 shows the deformation and stress distribution for regular honeycomb 7 mil paper under 5% compressive strain in the y direction.

Figure 4-11: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% compressive strain in the y direction.

4.1.3.4. Chevron

Based on the test conditions discussed in section 3.2.3.3.4, Figure 4-12 shows the deformation and stress distribution for chevron 7 mil paper under 5% compressive strain in the y direction.

Figure 4-12: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% compressive strain in the y direction.

4.1.4. Shear in the x direction

The materials used were introduced in section 3.3.1. The boundary and loading conditions were also introduced in section 3.2.3.

4.1.4.1. Inverted honeycomb

Based on the test conditions discussed in section 3.2.3.4.1, Figure 4-13 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% shear strain in the x direction.

Figure 4-13: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the x direction.

4.1.4.2. Waterbomb

Based on the test conditions discussed in section 3.2.3.4.2, Figure 4-14 shows the deformation and stress distribution for Waterbomb 7 mil paper under about 5% shear strain in the x direction.

Figure 4-14: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% shear strain in the x direction.

4.1.4.3. Regular honeycomb

Based on the test conditions discussed in section 3.2.3.4.3, Figure 4-15 shows the deformation and stress distribution for regular honeycomb 7 mil paper under 5% shear strain in the x direction.

Figure 4-15: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the x direction.

4.1.4.4. Chevron

Based on the test conditions discussed in section 3.2.3.4.4, Figure 4-16 shows the deformation and stress distribution for chevron 7 mil paper under about 5% shear strain in the x direction.

Figure 4-16: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% shear strain in the x direction.

4.1.5. Shear in the y direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were also described in section 3.2.3.5.

4.1.5.1. Inverted honeycomb

Based on the test conditions discussed in section 3.2.3.5.1, Figure 4-17 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% shear strain in the y direction.

Figure 4-17: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper inverted honeycomb model under 5% shear strain in the y direction.

4.1.5.2. Waterbomb

Based on the test conditions discussed in section 3.2.3.5.2, Figure 4-18 shows the deformation and stress distribution for Waterbomb 7 mil paper under 5% shear strain in the y direction.

Figure 4-18: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper Waterbomb model under 5% shear strain in the y direction.

4.1.5.3. Regular honeycomb

Based on the test conditions discussed in section 3.2.3.5.3, Figure 4-19 shows the deformation and stress distribution for inverted honeycomb 7 mil paper under 5% shear strain in the y direction.

Figure 4-19: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper regular honeycomb model under 5% shear strain in the y direction.

4.1.5.4. Chevron

Based on the test conditions discussed in section 3.2.3.5.4, Figure 4-20 shows the deformation and stress distribution for chevron 7 mil paper under 5% shear strain in the y direction. 60

Figure 4-20: Deformation and Von-Mises stress distribution of 0.18 mm thick Aramet® paper chevron model under 5% shear strain in the y direction.

4.2. Analysis and comparison

This work was done to find the normalized effective Young’s modulus and effective Poisson’s ratios of the cores for sandwich structures under compressive loads and normalized effective shear modulus under shear loads. The simulation results have been analyzed to find such values. Before going through analysis and comparison of the results for the models, we need to find how reliable these results are. Since the approach

employed for this study was based on the approach Heimbs [43], [56] used for his study on compressive and shear and he verified the numerical data with the results of experimental tests, if our results for these two models are verified by those of his work, we could validate the approach and extend the use of it for other models. In his work, the linear section of stress-strain curve shows that the level of stress for 1% compressive strain for regular honeycomb core inside the sandwich panel is 2 MPa. In our study the level of stress for 1% compressive strain is 2.1 MPa. For chevron the level of stress in

Heimbs’es work is 2.2 MPa and in our work it is 2.2 MPa. Also, the shear results of his both numerical and experimental studies show that the level of stress for chevron under shear is 0.7 MPa and the value of stress for chevron under shear in our work is 0.75 MPa.

This verifies that our compressive and shear results in our work for regular honeycomb and chevron are accurate. Having the results verified by literature gives us enough confidence to rely on the compressive and shear results for the other models, waterbomb and inverted honeycomb and it could be stated that the Comsol Multiphysics finite element models showed a high level of precision. In this section the geometries and loading conditions’ results will be compared to find the best geometry and loading for any application. The effective Young’s modulus and effective shear modulus were normalized based on the approach in [56] to the material’s Young’s modulus and relative density.

휌∗ 푅푒푙푎푡𝑖푣푒 퐷푒푛푠𝑖푡푦 = 휌푠 Where: 푚 휌∗ = 푚표푑푒푙 푣푚표푑푒푙 And

푚푝푎푝푒푟 휌푠 = 푣푝푎푝푒푟 Now

푚푝푎푝푒푟 = 푚푚표푑푒푙 Therefore:

푣 푅푒푙푎푡𝑖푣푒 퐷푒푛푠𝑖푡푦 = 푝푎푝푒푟=푡푎0푏0 푣푚표푑푒푙 ℎ푎푏

Where: ′ 푎0: 푃푎푝푒푟 푠 푙푒푛푔푡ℎ ′ 푏0: 푃푎푝푒푟 푠 푤𝑖푑푡ℎ 푡: 푃푎푝푒푟′푠 푡ℎ𝑖푐푘푛푒푠푠 푎 = 푀표푑푒푙′푠 푙푒푛푔푡ℎ 푏 = 푀표푑푒푙′푠 푤𝑖푑푡ℎ ℎ = 푀표푑푒푙′푠 ℎ푒𝑖푔ℎ푡 퐸푓푓푒푐푡𝑖푣푒 푌표푢푛푔′푠 푚표푑푢푙푢푠 푁표푟푚푎푙𝑖푧푒푑 퐸 = 푀푎푡푒푟𝑖푎푙′푠 푌표푢푛푔′푠 푚표푑푢푙푢푠 ∗ 푅푒푙푎푡𝑖푣푒 퐷푒푛푠𝑖푡푦

퐸푓푓푒푐푡𝑖푣푒 푠ℎ푒푎푟 푚표푑푢푙푢푠 푁표푟푚푎푙𝑖푧푒푑 퐺 = 푀푎푡푒푟𝑖푎푙′푠 푠ℎ푒푎푟 푚표푑푢푙푢푠 ∗ 푅푒푙푎푡𝑖푣푒 퐷푒푛푠𝑖푡푦

Table 4.1 shows the relative density and material’s Young’s and shear moduli for both models.

Table 4.1: Values used for normalizing the results

Model Relative Density Material’s Young’s Material’s Shear

Modulus (MPa) Modulus (MPa)

Inverted Honeycomb 0.01766 185 71

Waterbomb 0.01986 185 71

Regular Honeycomb 0.008575 185 71

Chevron 0.026879 185 71

The normalization was done to exclude the effect of type and amount of the used material for simulation on the results. The models are compared to each other based on two aspects of geometry and loading response.

4.2.1. Geometry

All geometries’ performances under discussed loading conditions are investigated in this section. The first studied geometry is inverted honeycomb. Its behaviors under different loading conditions are given in Tables 4.2 and 4.3 and Figures 4-11 and 4-12.

Table 4.2: Inverted honeycomb’s normalized effective Young’s modulus under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.9711211

x compression 0.0000375

y compression 0.0001618

Table 4.3: Inverted honeycomb’s normalized effective shear modulus under different shear loading conditions

Load Normalized effective shear modulus

x shear 0.30505494

y shear 0.33589268

Figure 4-21: Inverted honeycomb’s normalized effective Young’s modulus under different compression loading conditions

Figure 4-22: Inverted honeycomb’s normalized effective shear modulus under different shear loading conditions

Inverted honeycomb geometry demonstrates a much higher effective Young’s modulus in compression in the z direction. As it is seen in the chart and table, the normalized effective Young’s modulus in the x and y directions are negligible when compared with the normalized effective Young’s modulus in the z direction. In the case of shear, the effective shear modulus in the y direction is about 10% higher compared to the shear modulus in the x direction.

Waterbomb’s results are shown in Tables 4.4 and 4.5 and Figures 4-13 and 4-14.

Table 4.4: Waterbomb’s normalized effective Young’s modulus under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.0122180

x compression 0.0014833

y compression 0.0069655

Table 4.5: Waterbomb’s normalize effective shear modulus under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.0913397

y shear 0.2115857

Figure 4-23: Waterbomb’s normalize effective Young’s modulus under different compression loading conditions

Figure 4-24: Waterbomb’s normalize effective shear modulus under different shear loading conditions

Like inverted honeycomb, waterbomb shows a greater value for the normalized effective Young’s modulus in the z direction than its values for compression in the x and y directions. However, the values for compression in the x and y are not negligible and are almost 12% and 56% of that of compression in the z direction. The normalized effective shear modulus in the y direction is about 2.3 times normalized effective shear modulus in the x direction.

The results for compression and shear for regular honeycomb are shown in the

Figures 4-25 and 4-26 and tables 4-6 and 4-7.

Table 4.6: Regular honeycomb’s normalized effective Young’s modulus under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.999988682

x compression 0.000161305

y compression 0.000166334

Table 4.7: Regular honeycomb’s normalized effective shear modulus under different shear loading conditions

Load Normalized effective shear modulus

x shear 0.484553639

y shear 0.342759058

Normalized Effective Young's Modulus 1.2

0.8

0.6

0.4

0.2

0 Z compression X compression Y compression

Figure 4-25: Regular honeycomb’s normalized effective Young’s modulus under different compression loading conditions 69

Normalized Effective Shear Modulus 0.6

0.5

0.4

0.3

0.2

0.1

0 X shear Y shear

Figure 4-26: Regular honeycomb’s normalized effective shear modulus under different shear loading conditions

Like inverted honeycomb, regular honeycomb geometry demonstrates a much higher effective Young’s modulus in compression in the z direction. As it is seen in the diagram and table, the normalized effective Young’s modulus in the x and y directions are negligible when compared with the normalized effective Young’s modulus in the z direction. In the case of shear, the effective shear modulus in the x direction is about 20% higher compared to the shear modulus in the y direction.

Chevron’s results are shown in Tables 4.8 and 4.9 and Figures 4-27 and 4-28.

Table 4.8: Chevron’s normalized effective Young’s modulus under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.009831565

x compression 0.000139605

y compression 0.000242449

Table 4.9: Chevron’s normalize effective shear modulus under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.338532507

y shear 0.30895989

Normalized Effective Young's Modulus 0.012

0.01

0.008

0.006

0.004

0.002

0 Z compression X compression Y compression

Figure 4-27: Chevron’s normalize effective Young’s modulus under different compression loading conditions

Normalized Effective Shear Modulus 0.345 0.34 0.335 0.33 0.325 0.32 0.315 0.31 0.305 0.3 0.295 0.29 X shear Y shear

Figure 4-28: Chevron’s normalize effective shear modulus under different shear loading conditions

Like other models, chevron shows a greater value for the normalized effective

Young’s modulus in the z direction than its values for compression in the x and y directions. The normalized effective shear modulus in the x direction is about 1.1 times of normalized effective shear modulus in the y direction.

4.2.2. Loading Conditions

All loading conditions for different geometries are investigated in this section.

The first studied loading is compression in the z direction. All models’ behaviors in this loading condition are given in Table 4.10 and Figures 4-29, and 4-30.

Table 4.10: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the z direction.

Model Normalized effective Young’s 휈푥푧 휈푦푧

modulus

Waterbomb 0.012218 2.5815 1.1706

Inverted honeycomb 0.971121 0.2998 0.2999

Chevron 0.009831 8.301584 6.2999

Regular honeycomb 0.999988 0.3 0.3

Normalized Effective Young's Modulus 1.2

0.8

0.6

0.4

0.2

0 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron

Figure 4-29: Normalized effective Young’s modulus for all models for compression in the z direction.

Poisson's Ratios 9 8 7 6 5 4 3 2 1 0 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron

Poisson xz Poisson yz

Figure 4-30: Effective Poisson’s ratios xz and yz for all models for compression in the z direction.

In the case of compression in the z direction, as expected, the normalized effective Young’s modulus presented by regular and inverted honeycombs are 80-100 times greater than the values for waterbomb and chevron. All Poisson’s ratios are positive, and chevron shows the largest Poisson’s ratios than the other models.

Waterbomb’s Poisson’s ratios are less than chevron but still much bigger than honeycombs.

In the case of compression in the x direction, waterbomb demonstrates a much larger effective Young’s modulus than honeycombs and chevron. It is more than 10 times greater than the value for the chevron and 9 times larger than the value for regular honeycomb. Waterbomb and chevron have positive Poisson’s ratios zx and negative

Poisson’s ratios yx, whereas the regular and inverted honeycombs’ Poisson’s ratios zx are almost zero. Poisson’s ratio yx for inverted honeycomb is, as expected, negative and the one for regular honeycomb is positive. It means that auxeticity helps enhancing the

effective Young’s modulus in the case of compression in the x direction. Table 4.11 and

Figures 4-31, and 4- show the comparison of the models for compression in the x

direction.

Table 4.11: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the x direction.

Model Normalized effective Young’s 휈푧푥 휈푦푥

modulus

Waterbomb 0.001483 0.3136 -0.4332

Inverted honeycomb 0.000037 Almost zero -0.4998

Chevron 0.000139 0.1175 -0.7529

Regular honeycomb 0.000161 Almost zero 0.9996

Normalized Effective Young's Modulus 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 Regular Inverted Waterbomb Chevron Honeycomb Honeycomb

Figure 4-31: Normalized effective Young’s modulus for all models for compression in the x direction. 75

Poisson's Ratios 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron -0.4 -0.6 -0.8 -1

Poisson zx Poisson yx

Figure 4-32: Effective Poisson’s ratios zx and zy for all models for compression in the x direction.

Compression in the y direction results in different behaviors. Table 4.12 and

Figures 4-33, and 4-34 show those for different models.

Table 4.12: Different models’ normalized effective Young’s modulus and Poisson’s ratios under compression in the y direction.

Model Normalized effective 휈푧푦 휈푥푦

Young’s modulus

Waterbomb 0.006965 0.6874 -2.0241

Inverted honeycomb 0.000161 Almost zero -1.9993

Chevron 0.000242 0.1544 -1.3040

Regular honeycomb 0.000166 Almost zero 0.9996

Normalized Effective Young's Modulus 0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron

Figure 4-33: Normalized effective Young’s modulus for all models for compression in the y direction.

Poisson's Ratios 1.5 1 0.5 0 -0.5 Regular Honeycomb Inverted Waterbomb Chevron -1 Honeycomb -1.5 -2 -2.5

Poisson zy Poisson xy

Figure 4-34: Effective Poisson’s ratios zy and zx for all models for compression in the x direction.

In this case for a second time, waterbomb’s normalized effective Young’s modulus is so greater in comparison to the other models. Chevron also shows a 1.5 times greater value than honeycombs. Again, Poisson’s ratios zy for waterbomb and chevron are positive while the ones for the regular and inverted honeycombs are almost zero. All

models, but regular honeycomb, demonstrate negative Poisson’s ratios xy while waterbomb’s value is less than 2% larger than the inverted honeycomb’s.

Tables 4.13 and 4.14 and Figures 4-35 and 4-36 show shear results.

Table 4.13: Different models’ normalized effective shear modulus shear in the x direction

Model Normalized shear modulus

Waterbomb 0.091339

Inverted honeycomb 0.305054

Chevron 0.338532

Regular honeycomb 0.484553

Table 4.14: Different models’ normalized effective shear modulus shear in the y direction

Model Normalized shear modulus

Waterbomb 0.211585

Inverted honeycomb 0.335892

Chevron 0.338532

Regular honeycomb 0.342759

Normalized Effective Shear Modulus 0.6

0.5

0.4

0.3

0.2

0.1

0 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron

Figure 4-35: Normalized effective shear modulus of all models for shear in the x direction.

Normalized Effective Shear Modulus 0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 Regular Honeycomb Inverted Honeycomb Waterbomb Chevron

Figure 4-36: Normalized effective shear modulus of all models for shear in the y direction.

As Tables and Figures clearly show, regular honeycomb geometry provides larger shear moduli in both x and y directions. In the case of x shear, chevron provides 1.1 times 79

larger value than inverted honeycomb and waterbomb’s is less than one fifth of regular honeycomb and one third of inverted honeycomb and chevron. In the case of y shear, waterbomb’s shear modulus is about two third of inverted honeycomb and chevron and half of regular honeycomb. Inverted honeycomb and chevron demonstrate almost equal shear moduli in y direction.

The results for z compression show that the type of the adhesives make no difference in normalize effective Young’s moduli for each model. The reason for this is that because the force causing the applied strain to the top facesheet is perpendicular to the panels and the layer of the adhesive, the change of material would result in no difference in reaction forces applied to the bottom facesheet and the level of stress for the cell ribs. Then after applying normalization method the results would be the same as well.

For compression in the x and y directions, since the load resulting the compressive strain is along the direction of the facesheets and adhesives, the type of adhesive makes difference. The level of change is a result of level of difference between adhesive’s average Young’s moduli.

Both regular honeycomb and inverted honeycomb present much higher effective

Young’s moduli under z compression than waterbomb and chevron. This is mainly because that honeycombs cell walls are along the direction of applied load and all of those walls must experience compressive strain. For waterbomb and chevron the walls have angels with the direction of the applied displacement and the outcome is lower built- up stress for the models and thus lower effective Young’s moduli for the entire model.

Under x compression, honeycombs and waterbomb present higher stiffness because they have ribs along the direction of load. For regular honeycomb because the direction of the

side ribs is outward there would be more load required to first fold those ribs and then compress the walls along the direction of the load. For waterbomb also there is an interaction between elements at the lowest level of the geometry resisting against compression. Lack of this for inverted honeycomb combined with the inward direction of inverted ribs for inverted honeycomb are the reasons that inverted honeycomb experiences less stress and thus lower effective Young’s moduli than regular honeycomb and waterbomb. For compression in the y direction, there is no wall for either type of honeycomb along the direction of the load and the cells fold along the fold line on the side ribs much easier. While waterbomb is stiffer due to existence of the elements at the lowest level interacting with each other and building up more stress for this model which results in a higher effective Young’s modulus. For shear, waterbomb has the lowest effective shear moduli because it only has one node for each cell to rotate around it. The other model have edges attached to the bottom facesheet and they experience more load along the wall ribs which trying to rotate around and along the edges connected to the bottom facesheet and they also undergo more shear stress inside the cell walls.

Chapter 5

5. Compression and Shear for Sandwich Panels

In this chapter, all of the cores will be examined between two facesheets creating sandwich panels and the results will be presented and analyzed. Facesheets are made of

CFRP with the mechanical properties presented in Table 5.1. Two different adhesives are used for gluing facesheets to the cores. Scotchweld 9323 B/A and SA 80 are two adhesives that have frequently been used in real world for manufacturing sandwich structures. Both of these glues are very suitable for gluing CFRP to Kevlar papers. Both of these adhesives have very high debonding tensile and shear stresses (>600 MPa) and bearing in mind that the maximum stresses in our study are in order of 100 MPa, the glues do the same in adhesion and the only difference is in mechanical properties such as

Young’s modulus and density. Again, the results are presented in a way that all models’ data of compression tests come first followed by the data of shear tests.

Table 5.1: Mechanical properties of CFRP

Thickness Average Young’s Density ( 푘푔 ) Poisson’s 푚3 (mm) Modulus (MPa) Ratio (Supplied by (From [2]) Manufacturer)

CFRP 1 70000 1600 0.3

Table 5.2: Mechanical properties of adhesives

Name Thickness Average Young’s Modulus Density ( 푘푔 ) (Supplied 푚3 (mm) (MPa) (Supplied by by Manufacturer) Manufacturer)

Scotchweld 0.025 1798 1120

9323 B/A

SA 80 0.025 3050 1380

5.1. Results

5.1.1. Compression in the z direction

The materials used were introduced in section 3.2.1 and beginning of this chapter.

The boundary and loading conditions were also described in section 3.2.3.

Figures 5-1 through 5-8 show the test results of all models with different adhesives for the compression in the z direction.

Figure 5-1: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction.

Figure 5-2: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction.

Figure 5-3: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction.

Figure 5-4: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction.

Figure 5-5: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction.

Figure 5-6: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction.

Figure 5-7: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the z direction.

Figure 5-8: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the z direction.

4.1.2. Compression in the x direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were, also, introduced in section 3.2.3.

Figures 5-9 through 5-16 show the test results of all models with different adhesives for the compression in the x direction.

Figure 5-9: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction.

Figure 5-10: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction.

Figure 5-11: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction.

Figure 5-12: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction.

Figure 5-13: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction.

Figure 5-14: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction.

Figure 5-15: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the x direction.

Figure 5-16: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the x direction.

4.1.3. Compression in the y direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were also introduced in section 3.2.3.

Figures 5-17 through 5-24 show the test results of all models with different adhesives for the compression in the y direction.

Figure 5-17: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction.

100

Figure 5-18: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction.

101

Figure 5-19: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction.

102

Figure 5-20: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction.

103

Figure 5-21: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction.

104

Figure 5-22: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction.

105

Figure 5-23: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% compressive strain in the y direction.

106

Figure 5-24: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% compressive strain in the y direction.

4.1.4. Shear in the x direction

The materials used were introduced in section 3.3.1. The boundary and loading conditions were also introduced in section 3.2.3.

Figures 5-25 through 5-32 show the test results of all models with different adhesives for the shear in the x direction.

107

Figure 5-25: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction.

108

Figure 5-26: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction.

109

Figure 5-27: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction.

110

Figure 5-28: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction.

111

Figure 5-29: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction.

112

Figure 5-30: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the x direction.

113

Figure 5-31: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the x direction.

114

Figure 5-32: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 5 % shear strain in the x direction.

4.1.5. Shear in the y direction

The materials used were introduced in section 3.2.1. The boundary and loading conditions were also described in section 3.2.3.5.

Figures 5-33 through 5-40 show the test results of all models with different adhesives for the shear in the x direction.

115

Figure 5-33: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction.

116

Figure 5-34: Deformation and Von-Mises stress distribution of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction.

117

Figure 5-35: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction.

118

Figure 5-36: Deformation and Von-Mises stress distribution of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction.

119

Figure 5-37: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction.

120

Figure 5-38: Deformation and Von-Mises stress distribution of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction.

121

Figure 5-39: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under 1% shear strain in the y direction.

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Figure 5-40: Deformation and Von-Mises stress distribution of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under 1% shear strain in the y direction.

5.2. Analysis and comparison

Again the comparison is done based on two aspects of geometry and loading response.

5.2.1. Geometry

Behavior of all sandwich panels with different geometries under discussed loading conditions are investigated in this section. The first studied geometry is inverted 123

honeycomb. Its behaviors with two different adhesives under different loading conditions are given in Tables 5.1 through 5.4 and Figures 5-41 and 5-44.

Table 5.1: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 1.012822

x compression 312.747529

y compression 148.834531

Table 5.2: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.391763

y shear 0.450074

124

Table 5.3: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 1.012845

x compression 498.074696

y compression 237.277453

Table 5.4: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.395724

y shear 0.450221

125

Normalized Effective Young's Modulus 350

300

250

200

150

100

0 Z compression X compression Y compression

Figure5-41: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Normalized Effective Shear Modulus 0.46 0.45 0.44 0.43 0.42 0.41 0.4 0.39 0.38 0.37 0.36 X shear Y shear

Figure 5-42: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

126

Normalized Effective Young's Modulus 600

500

400

300

200

100

0 Z compression X compression Y compression

Figure5-43: Normalized effective Young’s modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Normalized Effective Shear Modulus 0.46 0.45 0.44 0.43 0.42 0.41 0.4 0.39 0.38 0.37 0.36 X shear Y shear

Figure 5-44: Normalized shear modulus of sandwich panel with an inverted honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

The sandwich structure with an inverted honeycomb geometry demonstrates a higher effective Young’s modulus in compression in the x direction. The effective

127

Young’s moduli in the y direction are 2 times less than the ones in the x direction for both adhesives, and the effective Young’s moduli in the z direction are negligible in comparison with those in x and y directions. In the case of shear, for both adhesives, the effective shear moduli in the y direction are equal and about 10% higher compared to the shear moduli in the x direction. This exactly happened in the core only analysis, confirming that the core geometry is dominant in the case of shear and results are independent of facesheets and adhesives.

As it is clearly shown in the tables and figures, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression which totally makes sense. However, in the case of compression in the x and y directions, the type of adhesive makes a remarkable change. A stronger adhesive increases the effective

Young’s modulus of model in x and y compressions. Here, SA80 adhesive makes the sandwich structure 1.6 times stronger than what Scotchweld 9323 B/A does.

Next geometry is regular honeycomb. Its behaviors with two different adhesives under different loads are given in Tables 5.5 through 5.8 and Figures 5-45 and 5-48.

Table 5.5: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 1.095656

x compression 642.609540

y compression 252.867966

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Table 5.6: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.522702

y shear 0.464080

Table 5.7: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 1.095679

x compression 1023.588328

y compression 402.718843

129

Table 5.8: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.523142

y shear 0.464244

Normalized Effective Young's Modulus 700

600

500

400

300

200

100

0 Z compression X compression Y compression

Figure5-45: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

130

Normalized Effective Shear Modulus 0.53 0.52 0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43 X shear Y shear

Figure 5-46: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Normalized Effective Young's Modulus 1200

1000

800

600

400

200

0 Z compression X compression Y compression

Figure5-47: Normalized effective Young’s modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

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Normalized Effective Shear Modulus 0.53 0.52 0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43 X shear Y shear

Figure 5-48: Normalized shear modulus of sandwich panel with a regular honeycomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Like the sandwich panel with the inverted honeycomb core, the sandwich structure with a regular honeycomb geometry demonstrates a higher effective Young’s modulus in compression in the x direction. The effective Young’s moduli in the y direction are more than 2.5 times less than the ones in the x direction for both adhesives, and the effective Young’s moduli in the z direction are, again, negligible in comparison with those in x and y directions. In the case of shear, for both adhesives, the effective shear moduli in the x direction are equal and about 15% higher compared to the shear moduli in the y direction. This is again like what happened in the core only analysis, and like inverted honeycomb confirms that the core geometry is dominant in the case of shear and results do not depend on facesheets and adhesives.

Again, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression. But, in the case of compression in the x and y directions, the type of adhesive makes a significant change. The stronger SA80 adhesive 132

makes the sandwich structure 1.6 times stronger than what Scotchweld 9323 B/A does.

This ratio was observed exactly the same for inverted honeycomb as well.

Next geometry is waterbomb. Its performance with two different adhesives under different loading conditions are given in Tables 5.9 through 5.12 and Figures 5-49 and 5-

52.

Table 5.9: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.166312

x compression 369.988656

y compression 369.970463

Table 5.10: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.091905

y shear 0.198405

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Table 5.11: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.166321

x compression 589.318030

y compression 589.299459

Table 5.12: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.091969

y shear 0.198624

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Normalized Effective Young's Modulus 400

350

300

250

200

150

100

0 Z compression X compression Y compression

Figure5-49: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Normalized Effective Shear Modulus 0.25

0.2

0.15

0.1

0.05

0 X shear Y shear

Figure 5-50: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

135

Normalized Effective Young's Modulus 700

600

500

400

300

200

100

0 Z compression X compression Y compression

Figure 5-51: Normalized effective Young’s modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Normalized Effective Shear Modulus 0.25

0.2

0.15

0.1

0.05

0 X shear Y shear

Figure 5-52: Normalized shear modulus of sandwich panel with a waterbomb core, CFRP facesheets and SA80 adhesive under different compression loading conditions

For a third time, the sandwich structure with a waterbomb geometry shows negligible effective Young’s modulus in compression in the z direction. The effective

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Young’s moduli in the x and y directions are equal for both adhesives. In the case of shear, for both adhesives, the effective shear moduli in the x direction are equal and less than half of those for the y direction. This is again like what happened in the core only analysis, and like honeycombs approves that the core geometry is governing in the case of shear and results do not depend on facesheets and adhesives.

Like previous models, although, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression, in the case of compression in the x and y directions, it makes a major modification.

Last geometry in this section is chevron. Its performance with two different adhesives under different loading conditions are given in Tables 5.13 through 5.16 and

Figures 5-53 and 5-56.

Table 5.13: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.786064

x compression 146.413416

y compression 146.447847

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Table 5.14: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.413409

y shear 0.389307

Table 5.15: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective Young’s modulus

z compression 0.786101

x compression 233.228327

y compression 233.262738

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Table 5.16: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Load Normalized effective shear modulus

x shear 0.413772

y shear 0.395088

Normalized Effective Young's Modulus 160

140

120

100

0 Z compression X compression Y compression

Figure5-53: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

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Normalized Effective Shear Modulus 0.42 0.415 0.41 0.405 0.4 0.395 0.39 0.385 0.38 0.375 X shear Y shear

Figure 5-54: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and Scotchweld 9323 B/A adhesive under different compression loading conditions

Normalized Effective Young's Modulus 250

200

150

100

0 Z compression X compression Y compression

Figure 5-55: Normalized effective Young’s modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions

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Normalized Effective Shear Modulus 0.42

0.415

0.41

0.405

0.4

0.395

0.39

0.385 X shear Y shear

Figure 5-56: Normalized shear modulus of sandwich panel with a chevron core, CFRP facesheets and SA80 adhesive under different compression loading conditions

Like waterbomb, the sandwich structure with a chevron geometry demonstrates negligible effective Young’s modulus in compression in the z direction, and the effective

Young’s moduli in the x and y directions are equal for both adhesives. In the case of shear, for both adhesives, the effective shear moduli in the x direction are about 5% higher than those in the y direction. This is again like what happened in the core only analysis, and like previous models agrees that the core geometry governs in the case of shear and results do not depend on facesheets and adhesives.

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5.2.2. Loading Conditions

All loading conditions for different geometries are investigated in this section.

The first studied loading is compression in the z direction. All models’ behaviors in this

loading condition are given in Table 5.17 and Figures 5-57, and 5-58.

Table 5.17: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the z direction.

Normalized Poisson xz Poisson yz

Effective Young's

Modulus

Regular Honeycomb 1.095656779 0.0001899 0.0001807

Scotchweld

Regular Honeycomb 1.095679298 0.0001192 0.0001134

SA80

Waterbomb 0.166312031 5.21E-05 0.0002203

Scotchweld

Waterbomb SA80 0.166321006 3.263E-05 0.0001385

Inverted Honeycomb 1.012822125 0.0001076 0.0002099

Scotchweld

Inverted Honeycomb 1.012845411 6.757E-05 0.0001318

SA80

Chevron Scotchweld 0.786064559 0.0001379 0.000846

Chevron SA80 0.786101563 8.659E-05 0.0005378

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Normalized Effective Young's Modulus 1.2 1 0.8 0.6 0.4 0.2 0

Figure 5-57: Normalized effective Young’s modulus for all models for compression in the z direction.

Poisson's Ratios 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0

Poisson xz Poisson yz

Figure 5-58: Effective Poisson’s ratios xz and yz for all models for compression in the z direction.

In the case of compression in the z direction, like what happened in the core only simulations, all Poisson’s ratios are positive. But, the type of adhesive has a remarkable 143

influence on the effective Poisson’s ratios. All Poisson’s ratios dropped by 30-40% when switching from Scotchweld 9323 B/A to SA80. As expected, the normalized effective

Young’s modulus does not depend on the type of adhesive in this loading and remains the same for all models when switching from one adhesive to the other. Effective Young’s modulus presented by regular and inverted honeycombs are more than 6 times greater than the value for waterbomb. However, chevron provides an effective Young’s modulus

71% and 77% of inverted honeycomb and regular honeycomb moduli, respectively.

In the case of compression in the x direction, all Poisson’s ratios become positive which shows that facesheets are governing the deformation. When switching from

Scotchweld 9323 B/A adhesive to SA80 Poisson’s ratios slightly drop but it is much lower compared to the drop in the case of compression in the z direction. Again, regular honeycomb demonstrates the highest effective Young’s modulus, but, waterbomb demonstrates an 18% larger effective Young’s modulus than inverted honeycomb. It means that auxeticity helps enhancing the effective Young’s modulus in the case of compression in the x direction. Also, there is about 1.6 times enhancement for the effective Young’s moduli for all models when switching from Scotchweld 9323 B/A to

SA80 adhesive. Table 5.18 and Figures 5-59, and 5-60 show the comparison of the models for compression in the x direction.

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Table 5.18: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the x direction.

Model Normalized Poisson zx Poisson yx

Effective Young's

Modulus

Regular 642.6095405 0.11023478 0.30010741

Honeycomb

Scotchweld

Regular 1023.588328 0.10961521 0.30006743

Honeycomb SA80

Waterbomb 369.9886565 0.06831292 0.29991521

Scotchweld

Waterbomb 589.3180302 0.06492537 0.29994653

SA80

Inverted 312.7475298 0.04013958 0.29399069

Honeycomb

Scotchweld

Inverted 498.0746969 0.03778714 0.29416343

Honeycomb SA80

Chevron 146.4134168 0.03154211 0.30015249

Scotchweld

Chevron SA80 233.2283272 0.0292868 0.30009441

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Normalized Effective Young's Modulus 1200 1000 800 600 400 200 0

Figure 5-59: Normalized effective Young’s modulus for all models for compression in the x direction.

Poisson's Ratios 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Poisson zx Poisson yx

Figure 5-60: Effective Poisson’s ratios zx and yx for all models for compression in the x direction.

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Next studied loading is compression in the y direction. All models’ behaviors in

this loading condition are given in Table 5.19 and Figures 5-61, and 5-62.

Table 5.19: Different models’ effective Young’s modulus and Poisson’s ratios under compression in the y direction.

Normalized Poisson zy Poisson xy

Effective Young's

Modulus

Regular 252.8679663 0.1055029 0.27860965

Honeycomb Scotchweld

Regular 402.7188434 0.1056215 0.27855697

Honeycomb SA80

Waterbomb 369.9704635 0.3557724 0.29991779

Scotchweld

Waterbomb SA80 589.2994598 0.3527692 0.29994153

Inverted 148.8345312 0.0697516 0.29533049

Honeycomb Scotchweld

Inverted 237.2774536 0.0679154 0.29545972

Honeycomb SA80

Chevron 146.4478475 0.0996369 0.30024073

Scotchweld

Chevron SA80 233.2627381 0.0978493 0.30015185

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Normalized Effective Young's Modulus 700 600 500 400 300 200 100 0

Figure 5-61: Normalized effective Young’s modulus for all models for compression in the y direction.

Poisson's Ratios 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Poisson zy Poisson xy

Figure 5-62: Effective Poisson’s ratios xy and zy for all models for compression in the z direction.

In the case of compression in the y direction, the normalized effective Young’s modulus presented by waterbomb is the greatest. Regular honeycomb demonstrates an effective Young’s modulus 32% less than waterbomb. Inverted honeycomb and chevron

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do almost the same and their moduli are about 40% of that for waterbomb. All Poisson’s ratios are positive. Like the compression in the x direction, there is a slight drop when switching from Scotchweld 9323 B/A to SA80.

Tables 5.20 and 5.21 and Figures 5-63 and 5-64 show shear results.

Table 5.20: Different models’ normalized effective shear modulus shear in the x direction

Normalized Effective Shear Modulus

Regular Honeycomb Scotchweld 0.52270251

Regular Honeycomb SA80 0.523142138

Waterbomb Scotchweld 0.091905183

Waterbomb SA80 0.091969986

Inverted Honeycomb Scotchweld 0.391763634

Inverted Honeycomb SA80 0.39572496

Chevron Scotchweld 0.413409022

Chevron SA80 0.413772674

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Table 5.21: Different models’ normalized effective shear modulus shear in the y direction

Normalized Effective Shear

Modulus

Regular Honeycomb Scotchweld 0.464080442

Regular Honeycomb SA80 0.464244235

Waterbomb Scotchweld 0.198405831

Waterbomb SA80 0.19862452

Inverted Honeycomb Scotchweld 0.450074461

Inverted Honeycomb SA80 0.450221818

Chevron Scotchweld 0.389307984

Chevron SA80 0.395088574

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Normalized Effective Shear Modulus 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 5-63: Normalized effective shear modulus of all models for shear in the x direction.

Normalized Effective Shear Modulus 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Figure 5-64: Normalized effective shear modulus of all models for shear in the y direction.

As Tables and Figures clearly show, the type of adhesive does not make any change in effective shear moduli. Regular honeycomb geometry provides the largest shear moduli in both x and y directions. However, inverted honeycomb presents almost

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equal value for effective shear modulus in y direction. Chevron does very well and demonstrate effective shear moduli of 79% and 85% of those for regular honeycomb in x and y directions, respectively. Its value in the x direction is even higher than inverted honeycomb.

Finally, Table 5.22 shows the best two models in each loading conditions. It helps choosing the best geometries for each loading for the future studies.

Table 5.22: The best two geometries for each loading

Model The Best The Second Best

Z Compression Regular Honeycomb Inverted Honeycomb

X Compression Regular Honeycomb Waterbomb

Y compression Waterbomb Regular Honeycomb

X Shear Regular Honeycomb Chevron

Y Shear Regular Honeycomb Inverted Honeycomb

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Chapter 6

6. Curvature for the Cores

In this phase, all of the cores will be examined under two different bending loadings. The goal is to find the curvatures created on the models under these loads and comparing them with each other. Models perform differently under these loads and each of them can be chosen for the appropriate application based on their behaviors. For both studies, all corners are fixed. In the following sections the results will be presented and an analysis and comparison on curvatures will be done.

6.l. Results

6.1.1. Distributed edge load

First loading condition is to apply a total of −10−2 N load in the z direction to the edges parallel to the x axis. Figure 6-1 shows how the load is applied to inverted honeycomb.

6.1.1.1. Inverted honeycomb

Based on the test conditions discussed in this chapter, Figures 6-2, 6-3 and 6-4 show the deformation, curvature and stress distribution for inverted honeycomb 7 mil paper under a total of −10−2 N load in the z direction to the edges parallel to the x axis.

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Figure 6-1: load is applied to the edges parallel to the x direction

Figure 6-2: xyz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load.

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Figure 6-3: yz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load.

Figure 6-4: zx view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load.

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6.1.1.2. Waterbomb

Based on the test conditions discussed in this chapter, Figures 6-5, 6-6 and 6-7 show the deformation, curvature and stress distribution for waterbomb 7 mil paper under a total of −10−2 N load in the z direction to the edges parallel to the x axis.

Figure 6-5: xyz view of deformation, curvature and stress distribution of inverted honeycomb under distributed edge load.

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Figure 6-6: yz view of deformation, curvature and stress distribution of waterbomb under distributed edge load.

Figure 6-7: zx view of deformation, curvature and stress distribution of waterbomb under distributed edge load.

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6.1.1.3. Regular honeycomb

Based on the test conditions discussed in this chapter, Figures 6-8, 6-9 and 6-10 show the deformation, curvature and stress distribution for regular honeycomb 7 mil paper under a total of −10−2 N load in the z direction to the edges parallel to the x axis.

Figure 6-8: xyz view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load.

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Figure 6-9: yz view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load.

Figure 6-10: zx view of deformation, curvature and stress distribution of regular honeycomb under distributed edge load.

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6.1.1.4. Chevron

Based on the test conditions discussed in this chapter, Figures 6-11, 6-12 and 6-13 show the deformation, curvature and stress distribution for chevron 7 mil paper under a total of −10−2 N load in the z direction to the edges parallel to the x axis.

Figure 6-11: xyz view of deformation, curvature and stress distribution of chevron under distributed edge load.

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Figure 6-12: yz view of deformation, curvature and stress distribution of chevron under distributed edge load.

Figure 6-13: zx view of deformation, curvature and stress distribution of chevron under distributed edge load.

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6.1.2. Concentrated center load

The other loading condition is a concentrated load of 10−2 N in the z direction to the center of the models.

6.1.2.1. Inverted honeycomb

Based on the test conditions discussed in this chapter, Figures 6-14, 6-15 and 6-16 show the deformation, curvature and stress distribution for inverted honeycomb 7 mil paper under a 10−2 N load in the z direction to the center of the model.

Figure 6-14: xyz view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load. 162

Figure 6-15: yz view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load.

Figure 6-16: zx view of deformation, curvature and stress distribution of inverted honeycomb under concentrated center load.

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6.1.2.2. Waterbomb

Based on the test conditions discussed in this chapter, Figures 6-17, 6-18 and 6-19 show the deformation, curvature and stress distribution for waterbomb 7 mil paper under a 10−2 N load in the z direction to the center of the model.

Figure 6-17: xyz view of deformation, curvature and stress distribution of waterbomb under concentrated center load.

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Figure 6-18: yz view of deformation, curvature and stress distribution of waterbomb under concentrated center load.

Figure 6-19: zx view of deformation, curvature and stress distribution of waterbomb under concentrated center load.

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6.1.2.3. Regular honeycomb

Based on the test conditions discussed in this chapter, Figures 6-20, 6-21 and 6-22 show the deformation, curvature and stress distribution for regular honeycomb 7 mil paper under a 10−2 N load in the z direction to the center of the model.

Figure 6-20: xyz view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load.

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Figure 6-21: yz view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load.

Figure 6-22: zx view of deformation, curvature and stress distribution of regular honeycomb under concentrated center load.

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6.1.2.4. Chevron

Based on the test conditions discussed in this chapter, Figures 6-23, 6-24 and 6-25 show the deformation, curvature and stress distribution for waterbomb 7 mil paper under a 10−2 N load in the z direction to the center of the model.

Figure 6-23: xyz view of deformation, curvature and stress distribution of chevron under concentrated center load.

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Figure 6-24: yz view of deformation, curvature and stress distribution of chevron under concentrated center load.

Figure 6-25: zx view of deformation, curvature and stress distribution of chevron under concentrated center load.

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6.2. Analysis and comparison

In this section the radius of curvature for edges and lines between midpoints are presented and compared with each other.

6.2.1. Distributed load

Under distributed edge load, models perform differently. Tables 6.1 – 6.4 show the radius of curvature for top, bottom, left and right edges and also the radius of curvature of the lines attaching midpoint of the left edge to the midpoint of the right edge

(called mid horizontal) and midpoint of the top edge to the midpoint of the bottom edge

(called mi vertical). The existence of negative values indicates that the model has shaped a saddle rather than a dome. The greater the absolute value of radii means that the edges are stiffer and tend more to stay flat.

Table 6.1: Radius of curvatures for inverted honeycomb under distributed edge load

Edge Radius of Curvature (m)

Bottom 0.99

Top 0.99

Left 1.64

Right 1.64

Mid Horizontal 1.87

Mid Vertical 6.66

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Table 6.2: Radius of curvatures for waterbomb under distributed edge load

Edge Radius of Curvature (m)

Bottom 1.02

Top 1.02

Left -5.36

Right -5.36

Mid Horizontal 1.19

Mid Vertical -3.03

Table 6.3: Radius of curvatures for regular honeycomb under distributed edge load

Edge Radius of Curvature (m)

Bottom 0.71

Top 0.71

Left -5

Right -5

Mid Horizontal 0.94

Mid Vertical -0.49

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Table 6.4: Radius of curvatures for chevron under distributed edge load

Edge Radius of Curvature (m)

Bottom 1.89

Top 1.89

Left 7.63

Right 7.63

Mid Horizontal 2.44

Mid Vertical -3.12

Table 6.1 clearly shows that inverted honeycomb creates a dome-shaped geometry after going under a bending caused by distributed load to the edges parallel to the x direction. Bottom and top edges bend more than left and right edges. The radius of curvature for bottom and top edges is 0.99 m which is about 60% of the radius of curvature for right and left edges. The mid horizontal line has a radius of curvature of

1.87 m which is even greater than the one for the edges parallel to the y axis. The value of 6.66 m for the radius of curvature for the mid vertical line shows that the model is much stiffer in the y direction against bending and stays almost flat, bearing in mind that the model dimensions are in order of 0.4 meters.

Although waterbomb creates a dome on bottom and top edges, table 6.2 shows that unlike inverted honeycomb, it demonstrates an anticlastic structure under this load.

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inverted honeycomb, stays dome-shaped, however, the mid vertical line shows anticlastic behavior. It is interesting that top and bottom edges bend almost the same for inverted honeycomb and waterbomb and the radii of curvatures are very close for them.

As expected, unlike inverted honeycomb, the regular honeycomb becomes saddle- shaped after the edge load is applied. The final radii of curvatures for edges are displayed in table 6.3. Bottom and top edges and mid horizontal line have positive values and left and right edges and mid vertical line have negative values confirming that the regular honeycomb has an anticlastic behavior. The curvatures for left and right edges have very large radii and are almost flat.

For chevron the load results in curves with a radius of 1.89 meters for top and bottom edges, although a value of 7.63 shows that the left and right edges stay almost flat. The mid horizontal line behaves as expected and follows the direction of bending, but the mid vertical line shows a little bit of anticlastic performance.

6.2.2. Concentrated load

Under concentrated center load, the synclastic and anticlastic behaviors of models are similar to those under distributed load. Tables 6.5 – 6.8 show the radius of curvature for top, bottom, left and right edges and also the radius of curvature of the lines of mid horizontal and mid vertical. Again the presence of negative values indicates that the model has shaped a saddle rather than a dome. The greater the absolute value of radii means that the edges are stiffer and tend more to stay flat.

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Table 6.5: Radius of curvatures for inverted honeycomb under concentrated center load

Edge Radius of Curvature (m)

Bottom 0.49

Top 0.49

Left 2.74

Right 2.74

Mid Horizontal 0.30

Mid Vertical 0.55

Table 6.6: Radius of curvatures for waterbomb under concentrated center load

Edge Radius of Curvature (m)

Bottom 0.90

Top 0.90

Left -6.45

Right -6.45

Mid Horizontal 1.26

Mid Vertical -2.07

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Table 6.7: Radius of curvatures for regular honeycomb under concentrated center load

Edge Radius of Curvature (m)

Bottom 0.57

Top 0.57

Left -2.41

Right -2.41

Mid Horizontal 0.73

Mid Vertical -0.41

Table 6.8: Radius of curvatures for chevron under concentrated center load

Edge Radius of Curvature (m)

Bottom 1.35

Top 1.35

Left 5.40

Right 5.40

Mid Horizontal 2.07

Mid Vertical -1.53

For concentrated load to the center, Inverted honeycomb, again, forms a synclastic structure. Like the edge loading condition, the left and right edges are stiffer than the top and bottom edges and have a radius of curvature of 2.74 meters which is very large when compared with the dimensions of the model. In comparison with the edge

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load, concentrated load causes curvier final shape because all of the radii of curvatures are lesser.

Table 6.6 shows the results for waterbomb under a concentrated load to the center. Negative values of right and left edges show that this geometry, again, performs in an anticlastic way and forms a saddle. However, these two edges have a very large radius of curvature which is too great compared with the initial dimensions. Like the previous study, mid horizontal and mid vertical lines have positive and negative radii of curvatures, respectively.

For regular honeycomb, for a second time, a saddle-shaped structure is the result of bending loading. The radii of curvature for bottom and top edges are very small showing that regular honeycomb is unable to maintain the initial shape under bending.

Again, mid horizontal value is positive and mid vertical line has a negative radius of curvature. Comparison of tables 6.7 and 6.3 shows that all radii of curvatures are larger in the case of edge load and the model is stiffer in such loading condition.

Lastly, chevron deforms like the previous loading condition, and forms a saddle- shaped geometry under a concentrated center load. Again, the right and left edges have a very large radius of curvature and one can say that it almost stays flat. Like regular honeycomb, the model is stiffer under distributed load to the edges.

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Chapter 7

7. Impact

Sandwich structures having comparatively thin facesheets are especially susceptible against impact loads. Velocities causing impact range from low velocity impacts (v < 10 m/s) to high velocity impacts (v < 1000 m/s) and hyper velocity impacts

(v > 1000 m/s) [43].

Although, facesheet material and thickness strongly affect the impact performance of sandwich panels, i.e. energy absorption capability, still, the core has considerable effect on the impact behavior. Also, some other factors like sandwich thickness, boundary conditions, impactor geometry and stiffness, influence the performance of the structure under impact. To assess the impact performance of sandwich structures in this research, low velocity impact simulation was performed based on an approach slightly different than the one employed in [43].

The impact tests in [43] were done by dropping a hemi- spherical steel impactor towards the top facesheet, the impactor has a diameter of 25.4 mm and a total weight of

1.56 kg. Once the impactor reaches the top plate it has a different velocity and thus a different kinetic energy for each test. Heimbs employed a hem-spherical impactor to further investigate the damage occurred to the cores. In our work instead of a hem- spherical impactor, a disk is employed to impact the sandwich panel. Also, instead of dropping the impactor which has an initial velocity and also gravity applies to it, the final

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kinetic energy of the impactor has been calculated and a constant velocity resulting in that kinetic energy is given to the disk impactor. The position of impact on the front facesheet was selected to be the center of the panel and the same location has been chosen in our work. However, there are studies which have proved that the location of contact between the impactor and he facesheet has negligible effects on the energy absorption capability of the core [43]. Boundary conditions are similar to Heimbs e.t. al.

[43] work in which the bottom plate was fixed to eliminate the effects of bending on absorbed energy of the models.

Last phase of this work involves investigation of the response of the models under impact. The goal of this section is to find the energy absorption capability of all four models under the impact energy of an impactor with five different kinetic energies.

Inverted honeycomb, waterbomb, regular honeycomb and chevron are each inserted between two facesheets to create sandwich panels with different core geometries. Each sandwich structure is then impacted by a disk with a constant velocity. Five data points for each structure are enough for us to find the percentage of kinetic energy of the impactor absorbed by the core and converted to internal energy by causing plastic deformation. The expectation is that a part of the impact energy is absorbed by the core; some of the energy is absorbed by the facesheet which is on the side that the impactor contacts the model and the rest goes back to the impactor causing it to bounce back. If impact energy does not reach to the bottom plate (the one far from the impactor), it means that it is isolated from the impact and the model is safe enough to be used as protection gears like helmets or part of aircraft fuselage or vehicles’ body or anywhere else that a high impact resistance is needed.

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Three out of four models are symmetric in the z direction, therefore the direction of impact does not matter. Waterbomb is the one which is not symmetric, so impact has been studied for this model in both z and –z directions.

Since the cores are made of Kevlar paper which has a relatively low stiffness and also this study is done to only compare the energy absorption capabilities of four different structures, the level of impact energies have been chosen to be low which does not make any difference in objectives of this work and will provide enough information for the comparison between models. Five different velocities result in 10 mJ, 20 mJ, 30 mJ, 40 mJ and 50 mJ. Boundary conditions are similar to Heimbs et. al. [43] work in which the bottom plate was fixed to eliminate the effects of bending on absorbed energy of the models.

Like all other parts of this work, first the results of the impact study are presented and next an analysis is conducted to compare the performance of them with each other.

7.1. Results

Impact transfers energy from the moving disk to the sandwich panel. The structure’s absorbed energy is that portion of the impact energy, which has caused plastic deformation in the core. It includes the strain energy of the core structure and the front facesheet and frictional losses in the contact area and at the boundaries of the plate.

Frictional losses at guide rails and air resistance during the impact event are usually expected to be small, but they are in fact indirectly included in the absorbed energy. The average of absorbed energy of a sandwich panel with a regular honeycomb in this work is about 60% which when compared with the experimental and numerical results of the tests conducted in [66] verifies that the results obtained in this study are accurate. After 179

confirming that the approach is realistic and one set of results are verified by other studies, it is possible to rely on other sets of results for other geometries, inverted honeycomb, chevron and waterbomb.

As mentioned earlier, part of the impact energy is absorbed by the top facesheet of the composite models. Figure 7-1 shows the absorbed energy of the top plate under an impact with 10 mJ. Since all models have similar plates with exact same materials and mechanical properties, this energy is the same for all models.

Figure 7-1: Absorbed energy by top plate under 10 mJ impact

For all models and all impact energies, top plate absorbs about 14% of the total energy.

Figures 7-2 – 7-6 show the stress distribution of the models under 10 mJ impact energy.

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Figure 7-2: Stress distribution and deformation of inverted honeycomb under 10 mJ impact

Figure 7-3: Stress distribution and deformation of waterbomb under 10 mJ impact

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Figure 7-4: Stress distribution and deformation of regular honeycomb under 10 mJ impact

Figure 7-5: Stress distribution and deformation of chevron under 10 mJ impact

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Figure 7-6: Stress distribution and deformation of upside down waterbomb under 10 mJ impact

Table 7.1: maximum stress in the cores.

Model Maximum stress (MPa)

Inverted Honeycomb 30.33

Waterbomb 31.35

Waterbomb (upside down) 33.45

Regular Honeycomb 27.97

Chevron 24.61

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Bearing in mind that the failure stress of the Kevlar paper is 14 MPa, all models go through plastic deformation under 10 mJ and definitely they do the same for higher energies.

Figure 7-7 – 7-11 and table 7.2 show the energy absorbed by the cores for 10 mJ impact energy.

Figure 7-7: Absorbed energy by inverted honeycomb under 10 mJ impact

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Figure 7-8: Absorbed energy by waterbomb under 10 mJ impact

Figure 7-9: Absorbed energy by waterbomb upside down under 10 mJ impact

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Figure 7-10: Absorbed energy by regular honeycomb under 10 mJ impact

Figure 7-11: Absorbed energy by chevron under 10 mJ impact

Table 7.2: Percentage of energy absorbed by the model under 10 mJ impact

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Model Absorbed Energy (%)

Inverted Honeycomb 78

Waterbomb 41

Waterbomb (upside down) 46

Regular Honeycomb 68

Chevron 33

Figures 7-12 – 7-16 and table 7.3 show the absorbed energy of the models under

20 mJ impact.

Figure 7-12: Absorbed energy by inverted honeycomb under 20 mJ impact

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Figure 7-13: Absorbed energy by waterbomb under 20 mJ impact

Figure 7-14: Absorbed energy by waterbomb upside down under 20 mJ impact

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Figure 7-15: Absorbed energy by regular honeycomb under 20 mJ impact

Figure 7-16: Absorbed energy by chevron under 20 mJ impact

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Table 7.3: Percentage of energy absorbed by the model under 20 mJ impact

Model Absorbed Energy (%)

Inverted Honeycomb 72

Waterbomb 35

Waterbomb (upside down) 37

Regular Honeycomb 60

Chevron 23

Figures 7-17 – 7-21 and table 7.4 show the absorbed energy of the models under

30 mJ impact.

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Figure 7-17: Absorbed energy by inverted honeycomb under 30 mJ impact

Figure 7-18: Absorbed energy by waterbomb under 30 mJ impact 191

Figure 7-19: Absorbed energy by waterbomb upside down under 30 mJ impact

Figure 7-20: Absorbed energy by regular honeycomb under 30 mJ impact 192

Figure 7-21: Absorbed energy by chevron under 30 mJ impact

Table 7.4: Percentage of energy absorbed by the model under 30 mJ impact

Model Absorbed Energy (%)

Inverted Honeycomb 60

Waterbomb 33

Waterbomb (upside down) 35

Regular Honeycomb 53

Chevron 20

Figures 7-22 – 7-26 and table 7.5 show the absorbed energy of the models under

40 mJ impact.

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Figure 7-22: Absorbed energy by inverted honeycomb under 40 mJ impact

Figure 7-23: Absorbed energy by waterbomb under 40 mJ impact

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Figure 7-24: Absorbed energy by waterbomb upside down under 40 mJ impact

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Figure 7-25: Absorbed energy by regular honeycomb under 40 mJ impact

Figure 7-26: Absorbed energy by chevron under 40 mJ impact

Table 7.5: Percentage of energy absorbed by the model under 40 mJ impact

Model Absorbed Energy (%)

Inverted Honeycomb 68

Waterbomb 35

Waterbomb (upside down) 35

Regular Honeycomb 50

Chevron 19

Figures 7-27 – 7-31 and table 7.6 show the absorbed energy of the models under

50 mJ impact.

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Figure 7-27: Absorbed energy by inverted honeycomb under 50 mJ impact

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Figure 7-28: Absorbed energy by waterbomb under 50 mJ impact

Figure 7-29: Absorbed energy by waterbomb upside down under 50 mJ impact

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Figure 7-30: Absorbed energy by regular honeycomb under 50 mJ impact

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Figure 7-31: Absorbed energy by chevron under 50 mJ impact

Table 7.6: Percentage of energy absorbed by the model under 50 mJ impact

Model Absorbed Energy (%)

Inverted Honeycomb 74

Waterbomb 56

Waterbomb (upside down) 50

Regular Honeycomb 52

Chevron 18

7.2. Analysis and Comparison

Tables 7.7 presents all results for all models and impact energies. Figure 7-32 – 7-

36 show the performance of each model in absorbing impact energies from 10 mJ to 50 mJ. Figure 7-37 compares all models’ performance under all impact loads.

Table 7.7: All results for impact

Model 10 20 30 40 50

mJ mJ mJ mJ mJ

Inverted 78 72 60 68 74

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Honeycomb

Waterbomb 41 35 33 35 56

Waterbomb 46 37 35 35 50

Upside down

Regular 68 60 53 50 52

Honeycomb

Chevron 33 23 20 19 18

Absorbed Energy (%) 90

0 0 1 2 3 4 5 6

Figure 7-32: Absorbed energy of inverted honeycomb under 10 mJ to 50 mJ impact

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Absorbed Energy (%) 60

0 0 1 2 3 4 5 6

Figure 7-33: Absorbed energy of waterbomb under 10 mJ to 50 mJ impact

Absorbed Energy (%) 60

0 0 1 2 3 4 5 6

Figure 7-34: Absorbed energy of waterbomb upside down under 10 mJ to 50 mJ impact 202

Absorbed Energy (%) 80

0 0 1 2 3 4 5 6

Figure 7-35: Absorbed energy of regular honeycomb under 10 mJ to 50 mJ impact

Absorbed Energy 35

0 0 1 2 3 4 5 6

Figure 7-36: Absorbed energy of chevron under 10 mJ to 50 mJ impact

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Absorbed Energy (%) 90

0 0 1 2 3 4 5 6

Inverted Honeycomb Waterbomb Waterbomb Upside down Regular Honeycomb Chevron

Figure 7-37: Absorbed energy of all models under 10 mJ to 50 mJ impact

As the energy increase from 10 mJ to 20 mJ, inverted honeycomb’s capability of absorbing energy decrease from 78% to 72% and for 30 mJ, inverted honeycomb absorbs

60% of the kinetic energy of the disk by plastic deformation. For 4o mJ and 50 mJ, inverted honeycomb is able to absorb 60 and 74 percent of impact energy, respectively.

Bearing in mind that the top facesheet absorbs about 14% of the energy, it means that a sandwich panel with a core of inverted honeycomb absorbs 74-88 percent of the impact energy and the rest goes back to the impactor, causing it to bounce back.

For waterbomb the energy absorbed by the core decreases from 41% to 35% as the impact energy goes up from 10 mJ to 20 mJ. It stays almost the same for 30mJ and 40 mJ while waterbomb shows a very good performance under 50mJ impact by absorbing more than 56% of the kinetic energy of the disk impactor. After all, waterbomb along

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with the top plate is able to absorb between 47 to 70 percent of the total impact energy leaving the bottom plate undamaged.

Upside down waterbomb performs almost the same as the waterbomb with an exception in the case of stress. The level of stress increases in this geometry faster and the core experiments the initiation of plastic deformation earlier.

Regular honeycomb, also, does not allow the impact energy pass the core and reaches to the bottom skin and like all other models, the level of stress and energy for the bottom facesheet remains at zero. For the core itself, the absorbed energy for all impact velocities is between 50-68 percent of the kinetic energy of the disk and when combined with the top plate, the whole model can kill 64 to 82% of total energy by plastic deformation.

Lastly, chevron absorbs 33 percent of 10mJ impact energy, but its ability of killing impact energy decreases to 23%, 20%, 19% and 18% for impacts with energies of

20mJ, 30 mJ, 40 mJ and 50 mJ, respectively. Along with the top facesheet, chevron is able to damp 32% to 47% of kinetic energy of the impactor.

Figure 7-37 and 7-38 clearly show that auxetic core of inverted honeycomb performs the best in impact with an ability of absorbing 70% of the impact energy in average. This confirms previous researchers’ statements that certain geometries with negative effective Poisson’s ratio outperform regular honeycomb that is the main core for sandwich structures in the market. The average of absorbed energy by regular honeycomb is 57% of total impact energy which is about 20% less than the one for inverted honeycomb. Waterbomb and upside down waterbomb absorb both absorb 40% of total energy in average which is still a good fraction of impact energy. However, they

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are remarkably weaker than both variations of honeycombs. The average of energy absorbed energy by chevron is about 23 percent making it the worst choice for using in structures where a higher impact resistance is required. After all, it is important to remember that on top of all of these numbers, the top plate absorbs about 14% of the impact energy and no energy is transferred to the bottom plate.

Absorbed Energy (%) 80 70 60 50 40 30 20 10 0 Inverted Waterbomb Waterbomb Regular Chevron Honeycomb Upside down Honeycomb

Figure 7-38: Average of absorbed energy by all models

These simulations are mainly valuable as a numerical assessment between different core structures in terms of energy plots. The energy curves are consistent, however, because of high complexity of the models the anticipation of an impeccable relationship would not be justifiable.

Energy absorption behavior of these models fall into two categories. Regular honeycomb and chevron show an almost linear reduction in energy absorption capability as the impact energy increases from 10 to 50 mJ. For inverted honeycomb and both waterbomb and upside-down waterbomb, the percentage of absorbed energy decrease as

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the impact energy increase from 10 mJ to 20 and 30 mJ. Then, they show a nonlinear growth in energy absorption ability as the impact energy increases to 40 and 50 mJ.

These simulations are mainly valuable as a numerical assessment between different core structures in terms of energy plots. Because of high complexity of the models the anticipation of an impeccable relationship would not be justifiable. Some changes particularly in the diagrams are unavoidable. This could be due to the failed elements which in some FE studies might be deleted and cracks will be generated. However, in the real world the failed parts still exist and interact with other parts of the model such as facesheets and impactor. The general relationship can be seen very reasonable which makes these models valuable for geometry studies and their extension for using other materials instead of Kevlar papers for making cores for the sandwich structures.

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Chapter 8

8. Conclusions and Future Work

In this study regular honeycomb has been selected as a benchmark and the performance of three other models- inverted honeycomb, chevron and waterbomb- under compressive, shear, bending and impact loading conditions were found and compared with the behavior of it. There are several studies in open literature about regular honeycomb and performance of this structure in different situations are well researched.

Therefore, verification of the results for regular honeycomb with the already known as accurate behavior of regular honeycomb provides enough confidence for relying on the approaches employed in this study and thus on the results obtained for other structures under such loading conditions. After all, it was found that inverted honeycomb is provides stiffness almost as well as regular honeycomb in case of compression in the z direction and shear in y direction. Waterbomb can be the primary choice when a higher stiffness in y direction is needed and it delivers a stiffness very close to the one for regular honeycomb in the case of compression in the x direction. Chevron is another foldcore which is easier to manufacture and provides shear stiffness in the x direction close enough to regular honeycomb to be considered to be chosen as an alternative after reviewing other factors like weight to strength ratio and manufacturing cost and ease.

Inverted honeycomb showed a very high impact energy absorption capability and might

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be of interest for further studies for different parameters of impact. Below, more details are provided about performance of these models under different loadings.

Four different core geometries which can be used as cores of sandwich structures were evaluated in this study. Inverted honeycomb, which is a modified version of well- known regular honeycomb cores was an interesting geometry to be studied in order to investigate its feasibility of being used as the core of sandwich panels. Waterbomb is a foldcore which, according to open literature, has not been investigated as a core for sandwich structures. Chevron is another foldcore which has raised interest among researchers, since it is an auxetic structure and might be a good alternative for existing core structures has also been studied in this work Also, regular honeycomb has been studied to be a good benchmark for comparison between models. These models were studied under 5 different loading conditions of compression in x, y and z directions and shear in x and y directions along with five different impact loads separately. Moreover, the performance of the models under two dissimilar bending loads causing curvatures in the models were investigated. This study was done to explore the effective Young’s modulus, effective Poisson’s ratios, and effective shear modulus, curvature under bending and energy absorption capability of the mentioned models.

To create the models, some software packages were employed. Waterbomb and chevron crease patterns were first created and converted to *.stl files in Rhinoceros 5 and finally imported to Comsol Multiphysics 5.1. Inverted honeycomb and regular honeycomb were built in Comsol Multiphysics 5.1. All simulations were then done in this finite element package, except for impact that has been studied in Abaqus.

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The physics employed for the models was shell and, therefore, all geometrical adjustments adequate for this physics were done. Some boundary conditions were common between all models and loading conditions, while some geometries and loading conditions demanded their own boundary conditions. The compression, shear and bending simulations were done based on linear elastic materials method because the levels of stress were so low. For impact the elastic perfectly plastic was the criteria. The material used for the simulation was Aramet® paper produced by Aramid, Ltd.

For core only analysis, inverted honeycomb demonstrates a considerable higher normalized effective Young’s modulus in compression in the z direction. Its normalized effective Young’s modulus in the y direction is five times greater than the one in the x direction, however, both are negligible when compared to that for z direction. In the case of shear, the normalized effective shear modulus in the y direction is about 10% higher compared to the shear modulus in the x direction. Waterbomb, again, shows a greater value of the normalized effective Young’s modulus in the z direction than its values for x and y directions. The normalized effective Young’s modulus for compression in the z direction is 8.7 and 1.7 times larger than its values for compression in the x and y directions respectively, while the effective shear modulus in the x direction is less than half of the effective shear modulus in the y direction. Like inverted honeycomb, regular honeycomb geometry demonstrates a much higher effective Young’s modulus in compression in the z direction. As it is seen in the diagram and table, the normalized effective Young’s modulus in the x and y directions are negligible when compared with the normalized effective Young’s modulus in the z direction. In the case of shear, the effective shear modulus in the x direction is about 20% higher compared to the shear

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modulus in the y direction. Like other models, chevron shows a greater value for the normalized effective Young’s modulus in the z direction than its values for compression in the x and y directions. The normalized effective shear modulus in the x direction is about 1.1 times of normalized effective shear modulus in the y direction. In the case of compression in the z direction, as expected, the normalized effective Young’s modulus presented by regular and inverted honeycombs are 80-100 times greater than the values for waterbomb and chevron. All Poisson’s ratios are positive, and chevron shows the largest Poisson’s ratios than the other models. Waterbomb’s Poisson’s ratios are less than chevron but still much bigger than honeycombs. In the case of compression in the x direction, waterbomb demonstrates a much larger effective Young’s modulus than honeycombs and chevron. It is more than 10 times greater than the value for the chevron and 9 times larger than the value for regular honeycomb. Waterbomb and chevron have positive Poisson’s ratios zx and negative Poisson’s ratios yx, whereas the regular and inverted honeycombs’ Poisson’s ratios zx are almost zero. Poisson’s ratio yx for inverted honeycomb is, as expected, negative and the one for regular honeycomb is positive. It means that auxeticity helps enhancing the effective Young’s modulus in the case of compression in the x direction. In the case of y compression, for a second time, waterbomb’s normalized effective Young’s modulus is so greater in comparison to the other models. Chevron also shows a 1.5 times greater value than honeycombs. Again,

Poisson’s ratios zy for waterbomb and chevron are positive while the ones for the regular and inverted honeycombs are almost zero. All models, but regular honeycomb, demonstrate negative Poisson’s ratios xy while waterbomb’s value is less than 2% larger than the inverted honeycomb’s. For shear, regular honeycomb geometry provides larger

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shear moduli in both x and y directions. In the case of x shear, chevron provides 1.1 times larger value than inverted honeycomb and waterbomb’s is less than one fifth of regular honeycomb and one third of inverted honeycomb and chevron. In the case of y shear, waterbomb’s shear modulus is about two third of inverted honeycomb and chevron and half of regular honeycomb. Inverted honeycomb and chevron demonstrate almost equal shear moduli in y direction.

For sandwich structures, the models were again supposed to the loads similar to those for core only analysis. Here the effects of two different adhesives for gluing facesheets to the cores were also explored. The sandwich structure with an inverted honeycomb geometry demonstrates a higher effective Young’s modulus in compression in the x direction. The effective Young’s moduli in the y direction are 2 times less than the ones in the x direction for both adhesives, and the effective Young’s moduli in the z direction are negligible in comparison with those in x and y directions. In the case of shear, for both adhesives, the effective shear moduli in the y direction are equal and about

10% higher compared to the shear moduli in the x direction. This exactly happened in the core only analysis, confirming that the core geometry is dominant in the case of shear and results are independent of facesheets and adhesives. The type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression which totally makes sense. However, in the case of compression in the x and y directions, the type of adhesive makes a remarkable change. A stronger adhesive increases the effective

Young’s modulus of model in x and y compressions. Here, SA80 adhesive makes the sandwich structure 1.6 times stronger than what Scotchweld 9323 B/A does. Like the sandwich panel with the inverted honeycomb core, the sandwich structure with a regular

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honeycomb geometry demonstrates a higher effective Young’s modulus in compression in the x direction. The effective Young’s moduli in the y direction are more than 2.5 times less than the ones in the x direction for both adhesives, and the effective Young’s moduli in the z direction are, again, negligible in comparison with those in x and y directions. In the case of shear, for both adhesives, the effective shear moduli in the x direction are equal and about 15% higher compared to the shear moduli in the y direction.

This is again like what happened in the core only analysis, and like inverted honeycomb confirms that the core geometry is dominant in the case of shear and results do not depend on facesheets and adhesives. Again, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression. But, in the case of compression in the x and y directions, the type of adhesive makes a significant change.

The stronger SA80 adhesive makes the sandwich structure 1.6 times stronger than what

Scotchweld 9323 B/A does. This ratio was observed exactly the same for inverted honeycomb as well. For a third time, the sandwich structure with a waterbomb geometry shows negligible effective Young’s modulus in compression in the z direction. The effective Young’s moduli in the x and y directions are equal for both adhesives. In the case of shear, for both adhesives, the effective shear moduli in the x direction are equal and less than half of those for the y direction. This is again like what happened in the core only analysis, and like honeycombs approves that the core geometry is governing in the case of shear and results do not depend on facesheets and adhesives. Like previous models, although, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression, in the case of compression in the x and y directions, it makes a major modification. Like waterbomb, the sandwich structure with a

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chevron geometry demonstrates negligible effective Young’s modulus in compression in the z direction, and the effective Young’s moduli in the x and y directions are equal for both adhesives. In the case of shear, for both adhesives, the effective shear moduli in the x direction are about 5% higher than those in the y direction. This is again like what happened in the core only analysis, and like previous models agrees that the core geometry governs in the case of shear and results do not depend on facesheets and adhesives. Like previous models, although, the type of adhesive does not affect the normalized effective Young’s modulus in the case of z compression, in the case of compression in the x and y directions, it makes a major modification.

In the case of compression in the z direction, like what happened in the core only simulations, all Poisson’s ratios are positive. But, the type of adhesive has a remarkable influence on the effective Poisson’s ratios. All Poisson’s ratios dropped by 30-40% when switching from Scotchweld 9323 B/A to SA80. As expected, the normalized effective

71% and 77% of inverted honeycomb and regular honeycomb moduli, respectively. In the case of compression in the x direction, all Poisson’s ratios become positive which shows that facesheets are governing the deformation. When switching from Scotchweld

9323 B/A adhesive to SA80 Poisson’s ratios slightly drop but it is much lower compared to the drop in the case of compression in the z direction. Again, regular honeycomb demonstrates the highest effective Young’s modulus, but, waterbomb demonstrates an

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18% larger effective Young’s modulus than inverted honeycomb. It means that auxeticity helps enhancing the effective Young’s modulus in the case of compression in the x direction. Also, there is about 1.6 times enhancement for the effective Young’s moduli for all models when switching from Scotchweld 9323 B/A to SA80 adhesive. In the case of compression in the y direction, the normalized effective Young’s modulus presented by waterbomb is the greatest. Regular honeycomb demonstrates an effective Young’s modulus 32% less than waterbomb. Inverted honeycomb and chevron do almost the same and their moduli are about 40% of that for waterbomb. All Poisson’s ratios are positive.

Like the compression in the x direction, there is a slight drop when switching from

Scotchweld 9323 B/A to SA80.

Again, the type of adhesive does not make any change in effective shear moduli.

Regular honeycomb geometry provides the largest shear moduli in both x and y directions. However, inverted honeycomb presents almost equal value for effective shear modulus in y direction. Chevron does very well and demonstrate effective shear moduli of 79% and 85% of those for regular honeycomb in x and y directions, respectively. Its value in the x direction is even higher than inverted honeycomb.

For curvature study, under distributed edge load, inverted honeycomb creates a dome-shaped geometry after going under a bending caused by distributed load to the edges parallel to the x direction. Bottom and top edges bend more than left and right edges. The radius of curvature for bottom and top edges is 0.99 m which is about 60% of the radius of curvature for right and left edges. The mid horizontal line has a radius of curvature of 1.87 m which is even greater than the one for the edges parallel to the y axis.

The value of 6.66 m for the radius of curvature for the mid vertical line shows that the

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model is much stiffer in the y direction against bending and stays almost flat, bearing in mind that the model dimensions are in order of 0.4 meters.

Although waterbomb creates a dome on bottom and top edges, table 6.2 shows that unlike inverted honeycomb, it demonstrates an anticlastic structure under this load.

The value of -5.36 show that when the load is applied to the top and bottom edges towards the negative z direction, the right and left edges tend to deform in the positive z direction and the final shape is a saddle-shaped geometry. The mid horizontal line, like inverted honeycomb, stays dome-shaped, however, the mid vertical line shows anticlastic behavior. It is interesting that top and bottom edges bend almost the same for inverted honeycomb and waterbomb and the radii of curvatures are very close for them.

For chevron the load results in curves with a radius of 0.48 meters for top and bottom edges, although a value of 25 shows that the left and right edges stay almost flat.

The mid horizontal line behaves as expected and follows the direction of bending, but the mid vertical line shows a little bit of anticlastic performance.

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large when compared with the dimensions of the model. In comparison with the edge load, concentrated load causes curvier final shape because all of the radii of curvatures are lesser.

For waterbomb, Negative values of right and left edges show that this geometry, again, performs in an anticlastic way and forms a saddle. However, these two edges have a very large radius of curvature which is too great compared with the initial dimensions.

Like the previous study, mid horizontal and mid vertical lines have positive and negative radii of curvatures, respectively.

In the case of impact, five different impact energies were applied to the models and it was shown that auxetic core of inverted honeycomb performs the best in impact with an ability of absorbing 70% of the impact energy in average. This confirms previous researchers’ statements that certain geometries with negative effective Poisson’s ratio outperform regular honeycomb that is the main core for sandwich structures in the

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market. The average of absorbed energy by regular honeycomb is 57% of total impact energy which is about 20% less than the one for inverted honeycomb. Waterbomb and upside down waterbomb absorb both absorb 40% of total energy in average which is still a good fraction of impact energy. However, they are remarkably weaker than both variations of honeycombs. The average of energy absorbed energy by chevron is about 23 percent making it the worst choice for using in structures where a higher impact resistance is required. After all, it is important to remember that on top of all of these numbers, the top plate absorbs about 14% of the impact energy and no energy is transferred to the bottom plate.

Future work could be the investigation of the effects of different boundary conditions on the behavior of the core only or composite models. Restricting edges from moving is an interesting boundary condition, since it is seen in many applications. Also, changing loading conditions like applying moment or torsion instead of compressive loads may result in different behaviors from the models. In addition, studying the performance of these models under combined loading could be an interesting research idea. For sandwich panels, changing the adhesive material with different mechanical properties would result in different behaviors. A very vast area of future work is investigating the performance of curved core only or sandwich structures under compression, shear or impact. Waterbomb has an ability of becoming tubular easily. A study of such structure in cylindrical coordinates under loads causing longitudinal stress or hoop stress would provide helpful results for different industries like aerospace, piping, etc.

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As it is shown in the curvature chapter of this study, model can become curved with a very low stress. Therefore, pre-stressed curved structures can be studied further for being employed where dome-shaped or saddle-shaped structures are needed. This could help manufacturers of ballistic or personal protection gears such as helmets, knee pads, and shoulder pads. Architecture and construction industry can also benefit these sandwich panels.

Also, instead of Kevlar papers, by employing stiffer materials and exploring the contribution of them in enhancing the effective Young’s moduli of the sandwich panels, it is possible to combine the benefits of auxeticity and higher stiffness. One major challenge here is that the concept of using foldcores is to take advantage of expansion and contraction of the models along fold lines. In other words, the fold lines function as hinges and let the model fold and unfold. Using much stiffer materials like carbon does not let the model to fold and unfold appropriately and the model might break under higher loads. So, rather than brittle materials, more ductile materials should be the first choice when thinking of getting higher stiffness.

Finally, sandwich panels in general and auxetic structures in particular have demonstrated desirable abilities in damping and isolating noise, vibration and harshness

(NVH). Currently, many industries suffer from NVH dampers which are so heavy and not very effective. For example, in automotive industry, structure-borne and air-borne

NVH is the main reasons of uncomfortable drive and ride for drivers and riders. Engine, road, and wind are the key producers of the noise and vibration and the NVH generated of two of which (engine and road) can be suppressed very effectively by using sandwich panels with auxetic cores. Cross car beams, brakes, seats, steering wheel columns, doors,

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floor, oil pan and driving shaft are the parts which can benefit sandwich panels with core structures with negative effective Poisson’s ratios. Conducting research on these structures and having it commercialized may lead to making vehicles quieter, lighter, safer and more comfortable.

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