3D-PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMBS: DESIGN, MECHANICAL PROPERTIES, AND DEFORMATION MECHANISMS By Chi Zhan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Civil Engineering ‒ Doctor of Philosophy 2022 ABSTRACT Lightweight materials and structures with extraordinary mechanical properties are desired for automotive, aerospace, and biomedical applications. Conventional lightweight materials such as foams have low density, large deformation and adjustable yield strength, and thus they are widely used in engineering applications. However, the mechanical performance is still far from satisfactory, due to the bending-dominated deformation mechanism determined by the stochastic microstructures. Inspired by nature, a novel type of lightweight materials, namely the hierarchical re-entrant honeycombs (H-ReHs) has been developed by replacing the traditional cell walls of re-entrant honeycombs by a second-order triangular hierarchy. The H-ReHs comprise fully controllable cellular structures ranging from microscale to macroscale. In addition, the geometry and pattern of the structures are precisely controlled via the emerging additive manufacturing techniques. We hypothesize that the H-ReHs outperform conventional lightweight materials and structures due to the microstructure of the material system, the unique interaction between the structural hierarchies, and the negative Poisson’s ratio effect. To test this hypothesis, H-ReHs with different geometric designs and printing materials have been manufactured. The mechanical performance of the H- ReHs has been fully characterized by quasi-static compression tests, dynamic impact tests, and quasi-static oblique tests. To reveal the deformation mechanisms of the H-ReHs, high-speed camera and 3D-digital image correlation have been utilized to analyze the internal strain field of the specimens under various loading conditions. It has been demonstrated that the 3D-printed H-ReHs exhibit enhanced specific stiffness, specific initial-buckling strength, structural stability, and specific energy absorption capacity in comparison to regular re-entrant honeycombs (R-ReHs). The H-ReHs exhibit stretching- dominated elastic behavior due to the second-order triangular hierarchy. The post-buckling behavior of the H-ReHs is more stable than the R-ReHs under various strain rates due to the interaction between the first and the second structural hierarchies. These findings provide insights into the design of next-generation lightweight yet mechanically robust materials and structures for future applications and mitigate energy crisis. 1 Copyright by CHI ZHAN 2022 Dedicated to my dearest family and my beloved wife v ACKNOWLEDGMENTS My time in Michigan State University has been the most joyful, challenging and meaningful period of my life so far. I couldn’t come along this unique journey without the support from many individuals that I am fortunate to have. First, I would like to express my deepest gratitude to my advisor, Prof. Weiyi Lu, for the unparallel support, guidance, patience, and encouragement throughout my four years study at MSU. I am grateful for the training and resources provided by him, especially the constant supports during the COVID-19 pandemic. I am also indebted for his significant efforts in refining my presentations and manuscripts. His innovative conceptualizations, open mindedness, unique perspectives towards challenges, and enthusiasm for the science inspire me to grow and become a better researcher and engineer. I am lucky and truly thankful to have Weiyi as my advisor during my PhD journey. I would also like to thank my great doctoral committee members I am lucky to have. I would like to thank Prof. Sara Roccabianca, for her valuable suggestions and inspirations on my research, which broaden my perspective and open up more possibility. I would like to thank Prof. Nizar Lajnef, for the constructive advice and questions, which helps me to build the grand picture of my research and expand my horizon on my ways of thinking. I would like to thank Prof. Bei Fan, for her always active and quick replies to my questions, and valuable feedback, comments, and suggestions on my research. I am sincerely thankful to my colleagues and friends, Dr. Mingzhe Li, Fuming Yang, for helps and teamwork together. Special thanks to Dr. Mingzhe Li, for strict instructions and kind suggestions throughout my PhD study. I want to thank Brian Wright, for the awesome technical vi supports on the 3D printing of testing specimens. I want to thank Prof. Shiwang Cheng, and his student, Ruikun Sun and Jie Yang, for the access to rheometers and helps. I would like to thank Prof. Guo Yang and Prof. Patrick Kwon, and their student, Jisheng Chen, Ryan Khawarizmi, and Juan Sandoval for the collaboration and helps. I would like to thank Prof. Wei Zhang and his student Jianzhou He for the training and access to the zetasizer. Besides, I also want to thank my friends, Xing Lu, Mamoon Shaafaey, Jun Guo, Yiyi Chu, Zheng Li, Lijiang Xu, Xuyang Li, Yang Chen, Hao Dong, Huan Zhang, Wenqi Li, thank for the friendship, helps, and joys on my research and life. I also want to thank my collaborators at Ford Motor Company, Dr. Robert McCoy, Linda Zhao, and Ahmad Rezaei Mojdehi, for their helps and supports. Special thanks to the financial support from Michigan State University (start-up fund, and TSGTD grant), and Ford Motor Company (MSU-Ford Alliance program). I am also grateful for receiving the Engineering Distinguished Fellowship, summer research fellowships, and travel fellowships provided by the College of Engineering, Department of Civil and Environmental Engineering, and the Graduate School of Michigan State University. Last but not least, I would like to thank my dearest family. To my parents, Guangkuan Zhan and Xiao Zhang, and my grandparents, Guangxin Zhang and Chisheng Zheng, thank you for everything you put to my upbringing, and for your always love, care, encouragement and support. I miss you all. I also owe my deep gratitude to my wife, Jing Zheng, I feel truly lucky to have you by my side, thank you for your unwavering support and always believe in me. vii TABLE OF CONTENTS LIST OF TABLES......................................................................................................................... x LIST OF FIGURES ..................................................................................................................... xi CHAPTER 1. MOTIVATION ................................................................................................... 1 CHAPTER 2. INTRODUCTION ............................................................................................. 8 2.1 Background ..................................................................................................................... 8 2.2 Research Objectives ...................................................................................................... 25 2.3 Dissertation Outline ...................................................................................................... 26 CHAPTER 3. 3D-PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB: ENHANCED MECHANICAL PROPERTIES AND THE UNDERLYING DEFORMATION MECHANISMS........................................................................................................................... 27 3.1 Introduction ................................................................................................................... 27 3.2 Materials and Methods .................................................................................................. 30 3.3 Results and Discussion ................................................................................................. 36 3.4 Conclusions ................................................................................................................... 56 CHAPTER 4. ANGLE EFFECT ON THE MECHANICAL PROPERTIES OF 3D- PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB ............................................ 58 4.1 Introduction ................................................................................................................... 58 4.2 Materials and Methods .................................................................................................. 61 4.3 Results and Discussion ................................................................................................. 65 4.4 Conclusions ................................................................................................................... 82 CHAPTER 5. STRAIN RATE EFFECT ON THE MECHANICAL PROPERTIES OF 3D- PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB ............................................ 84 5.1 Introduction ................................................................................................................... 85 5.2 Materials and Methods .................................................................................................. 87 5.3 Results and Discussion ................................................................................................. 91 5.4 Conclusion .................................................................................................................. 102 CHAPTER 6. OBLIQUE LOADING EFFECT ON THE CRUSHING BEHAVIORS OF HIERARCHICAL RE-ENTRANT HONEYCOMBS MADE WITH SOFT AND RIGID POLYMERS .............................................................................................................................. 104 6.1 Introduction ................................................................................................................. 105 6.2 Materials and Methods ................................................................................................ 107 6.3 Results and Discussion ................................................................................................112 viii 6.4 Conclusions ................................................................................................................. 127 CHAPTER 7. CONCLUSIONS AND FUTURE WORK ................................................... 130 7.1 Conclusions ................................................................................................................. 130 7.2 Future Work ................................................................................................................ 132 BIBLIOGRAPHY ..................................................................................................................... 135 APPENDIX ................................................................................................................................ 153 ix LIST OF TABLES Table 2.1. A summary of specifications of polymer 3D printing methods. .................................. 19 Table 3.1. The characteristics of the designed R- and H-ReH samples. ....................................... 32 Table 3.2. The scaling law parameters of relative stiffness and relative initial-buckling strength of ReHs versus relative density. ........................................................................................................ 56 Table 4.1. The characteristics of the designed H-ReH and R-ReH samples (𝜌 =0.34) with different angles. ........................................................................................................................................... 64 Table 4.2. Theoretical and experimental elastic modulus of R-ReH with different angles. ......... 74 Table 5.1. Characteristics of designed H-ReH and R-ReH with different relative densities. ....... 88 Table 5.2. Dynamic elastic modulus (Ed) and strength (σd) of H-ReHs and R-ReHs with various relative densities............................................................................................................................ 93 Table 5.3. Quasi-static elastic modulus (Eq) and strength (σq) of H-ReHs and R-ReHs with various relative densities............................................................................................................................ 94 Table 5.4. Dynamic mean crushing stress (σmd) and quasi-static mean crushing stress (σmq) at ε = 40% of H-ReHs. ............................................................................................................................ 97 Table 5.5. Elastic strain rate sensitivity (Se) and post-buckling strain rate sensitivity (Sp) of H- ReHs.............................................................................................................................................. 97 Table 6.1. The characteristics of designed H-ReHs and R-ReHs with different cell-wall angles. ......................................................................................................................................... 108 x LIST OF FIGURES Figure 1.1. Lightweight materials and structures used in ground and air vehicle body structures [8, 9]. .................................................................................................................................................... 2 Figure 1.2. Conventional lightweight materials with stochastic microstructures (a) Polyurethane foam [22]; (b) Microstructures of polyurethane foam [23]; (c) Aerogel monolith [24]; and (d) Syndiotactic polystyrene aerogel [25]. ........................................................................................... 3 Figure 1.3. Ashby map of strength versus density of the most common materials. (Reprinted from [26] with permission from John Wiley & Sons, Copyright 2018). ................................................. 4 Figure 1.4. Hierarchical materials found in nature. (a) Humerus bone structure of mice (Reprinted from [28] with permission from Oxford University Press, Copyright 2008); (b) Wing structure of Morpho rhetenor butterfly (Reprinted from [29] with permission from Springer Nature, Copyright 2003); (c) Beak structure of hornbill bird (Reprinted from [27] with permission from Annual Reviews, Copyright 2016). ............................................................................................................. 5 Figure 1.5. Architectures have been applied to structures to increase the structural efficiency. (a) Primitive wheel [30]; (b) Modern wheel [31]; (c) The pyramids of Giza [32]; (d) Eiffel Tower [33]. ...................................................................................................................................... 6 Figure 2.1. Honeycombs and its various applications. (a) Honeycomb lattice structure. (b) 3D- printed honeycomb in a cross-section of the airplane wing. (c) Honeycomb applied in shoe insole. (d) Honeycomb structured automobile tire [41]. ............................................................................ 9 Figure 2.2. (a) The scaling relationship between mechanical properties and density illustrating (b) stretching-dominated and (c) bending-dominated behavior exemplified by octet, honeycomb and metal foam. (Reprinted from [27] with permission from Annual Reviews, Copyright 2016)...... 11 Figure 2.3. Hierarchical structures of natural materials. (a) Bone (Reprinted from [49] with permission from Springer Nature, Copyright 2009); (b) Rattan wood (Reprinted from [50] with permission from Royal Society of Chemistry, Copyright 2009); (c) Moss (Sphagnum squarrosum) (Reprinted from [51] with permission from Royal Society, Copyright 2009); (d) Bamboo (Reprinted from [52] with permission from John Wiley & Sons, Copyright 2017); (e) Tendon [53].............................................................................................................................. 12 Figure 2.4. Examples of hierarchical architected materials simultaneously have ultralight, strong, and energy absorbing properties. (a) Hierarchical octahedra lattice (Reprinted from [59] with permission from National Academy of Sciences, Copyright 2015). (b) Ceramic octet-truss nanolattice (Reprinted from [60] with permission from The American Association for the xi Advancement of Science, Copyright 2014). (c) Metallic periodic hollow-tube microlattices (Reprinted from [61] with permission from The American Association for the Advancement of Science, Copyright 2011). ............................................................................................................. 14 Figure 2.5. (a) 2D re-entrant honeycomb; (b) 3D re-entrant honeycomb lattice; (c) 3D re-entrant honeycomb unit cell (Reprinted from [96] with permission from Elsevier, Copyright 2015). .... 15 Figure 2.6. Hierarchical structures incorporated in the re-entrant honeycombs. (a) star-shape [104]. (b) Vertex-based (Reprinted from [102] with permission from Elsevier, Copyright 2021). (c) Bidirectional hierarchy (Reprinted from [105] with permission from Elsevier, Copyright 2022). (d) Hexagon and triangle hierarchy (Reprinted from [106] with permission from Elsevier, Copyright 2019). ............................................................................................................................................ 17 Figure 2.7. FDM 3D printing technology and samples. (a) FDM system [124]. (b) FDM printer (Makerbot replicator 5th generation) in MSU library. (c) FDM printed hierarchical honeycomb using PLA. (d) FDM printed octet using ABS. ............................................................................. 20 Figure 2.8. Material jetting (Polyjet) printing technology and samples. (a) Schematic of material jetting (Polyjet) system [125]. (b) Polyjet printer (Objet Connex350) in Department of ECE in MSU. (c) Polyjet printed advanced Octet lattice and hierarchical re-entrant honeycomb using Rigur (thinnest wall is 0.5 mm) [126]. .................................................................................................... 22 Figure 2.9. (a) Aramis V8 3D-DIC system; (b) DIC system coupled with universal test machine during mechanical testing. ............................................................................................................ 24 Figure 3.1. Design and configuration of (a) R- and (b) H-ReH representative volume elements. ....................................................................................................................................... 30 Figure 3.2. 3D-printed standardized specimen and ReH samples with different relative densities. (a) Standardized dog bone specimen and cylindrical specimen for uniaxial tensile and compressive tests, respectively. (b-d) R-ReH specimens (ρ = 0.22, 0.32, and 0.48), and (e-g) H-ReH specimens (ρ = 0.23, 0.34, and 0.44). The central RVE of each sample (highlighted) represents the DIC field of view. .......................................................................................................................................... 32 Figure 3.3. Standardized testing results of the printing material: (a) uniaxial tensile test and (b) uniaxial compressive test. ............................................................................................................. 35 Figure 3.4. Mesh sensitivity analyses for (a) R-ReHs and (b) H-ReHs. ....................................... 35 Figure 3.5. Quasi-static compressive response of R- and H-ReHs with different relative densities: stress-strain curves. ....................................................................................................................... 36 xii Figure 3.6. Quasi-static compressive response of R- and H-ReHs with different relative densities. (a-c) Normalized stress-strain curves of R- and H-ReHs: (a) 𝜌 = 0.23, (b) 𝜌 = 0.34, and (c) 𝜌 = 0.44, (d) Specific stiffness, (e) Specific initial-buckling strength, and (f) Termination strain (D and F represent the termination mode of the ReHs is either densification or fracture). ...................... 38 Figure 3.7. (a) Specific energy absorption and (b) Mean crushing force of R- and H-ReHs. ...... 39 Figure 3.8. Experimental images of (a) R-ReH (𝜌 = 0.32) and (b) H-ReH (𝜌 = 0.34) at ε = 0.05, and (c) Poisson’s ratio (νxy) for corresponding R-ReH and H-ReH. ............................................. 41 Figure 3.9. DIC strain fields of R-ReH (𝜌 = 0.32) at ε = 0.05 (a-c) in 0°, 60° and – 60° directions, and H-ReH (𝜌 = 0.34) at ε = 0.04 (d-f) in 0°, 60° and – 60° directions. .................................... 42 Figure 3.10. Deformation patterns of (a) R-ReH (𝜌 = 0.32), (b) R-ReH (𝜌 = 0.48) and (c) H- ReH (𝜌 = 0.23). DIC strain fields of central RVEs of (d) R-ReH (𝜌 = 0.32) and (e) H-ReH (𝜌 = 0.23). ................................................................................................................................... 44 Figure 3.11. Effect of the slenderness of the 2nd order triangular hierarchy on (a) – (c) the local deformation mechanisms at ε = 0.1, where 𝑙ℎ/𝑡𝑡 = 8.64, 5.54, and 4.04; and (d) specific energy absorption of the H-ReHs. ............................................................................................................ 46 Figure 3.12. Deformation patterns of (a) R-ReH (𝜌 = 0.22), (b) H-ReH (𝜌 = 0.34) and (c) H- ReH (𝜌 = 0.44). .......................................................................................................................... 48 Figure 3.13. DIC strain fields of H-ReH (𝜌 = 0.23) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌 = 0.22) at ε = 0.04 (c) in 0°, and (d) 90°................................................................................. 49 Figure 3.14. DIC strain fields of H-ReH (𝜌 = 0.34) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌 = 0.32) at ε = 0.04 (c) in 0°, and (d) 90°................................................................................. 50 Figure 3.15. DIC strain fields of H-ReH (𝜌 = 0.44) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌 = 0.48) at ε = 0.04 (c) in 0°, and (d) 90°................................................................................. 51 Figure 3.16. DIC strain fields of R-ReH (𝜌 = 0.32) at ε = 0.01 (a-c) in 0°, + 60° and – 60° directions, and H-ReH (𝜌 = 0.34) at ε = 0.01 (d-f) in 0°, + 60° and – 60° directions................ 52 Figure 3.17. Numerical simulation of R- and H-ReHs. (a) Simulated constitutive behavior compared with experimental results, and simulated deformation patterns of (b) H-ReH, and (c) R-ReH. ..................................................................................................................................... 53 xiii Figure 3.18. Simulated deformation patterns of H- and R-ReHs. (a) H-ReH (𝜌 = 0.34), (b) H- ReH (𝜌 = 0.44), (c) R-ReH (𝜌 = 0.32), and (d) R-ReH (𝜌 = 0.48). ...................................... 54 Figure 3.19. Scaling laws for ReHs (a) Relative stiffness, and (b) Relative initial-buckling strength. ......................................................................................................................................... 56 Figure 4.1. Design and configuration of (a) 45°, (b) 60° and (c) 75° H-ReH and R-ReH (𝜌 =0.34) representative volume elements. ................................................................................................... 62 Figure 4.2. (a) Constitutive behavior of the Rigur printing material. (b-d) 3D-printed H-ReH and R-ReH samples with different angles: (b) 45°, (c) 60°, and (d) 75°. ............................................ 64 Figure 4.3. Quasi-static compressive response of R- and H-ReHs with three different cell-wall angle designs: 75ᵒ, 60ᵒ and 45ᵒ...................................................................................................... 66 Figure 4.4. The negative Poisson’s ratio effect of R- and H-ReHs with different cell-wall angles at ε=0.05. (a) Poisson’s ratio (νxy) versus cell-wall angles. Experimental images of ReHs: (b) 45º, (c) 60º, and (d) 75º.............................................................................................................................. 68 Figure 4.5. Initial elastic behaviors of H-ReHs and R-ReHs with different angles. (a) Elastic modulus, and (b) Strength. DIC strain field in inclined members: (c) alone inclined member axial direction and (d) alone horizontal direction. ................................................................................. 71 Figure 4.6. (a) R-ReH structure under compressive loading, and (b) free body diagram of the inclined member in R-ReH under loading. ................................................................................... 73 Figure 4.7. Post-buckling behavior of R-ReHs with different angles. (a) stress-strain curves, (b) termination strain, and (c-e) deformation pattern of R-ReHs at various compressive stages: (c) 45°, (d) 60°, and (e) 75°. ...................................................................................................................... 76 Figure 4.8. Post-buckling behavior of H-ReHs with different angles. (a) stress-strain curves, (b) termination strain, and (c-e) deformation pattern of H-ReHs at various compressive stages: (c) 45°, (d) 60°, and (e) 75°. U means undeformed stage. ......................................................................... 80 Figure 4.9. (a) Specific energy absorption (Es) of R and H-ReHs with different angles. (b) Maximum specific energy absorption (𝐸𝑠𝑚𝑎𝑥 ) of R and H-ReHs as a function of the cell-wall angle. ............................................................................................................................................. 82 Figure 5.1. Design of H-ReH and R-ReH with different relative densities. (a) 𝜌 = 0.17, (b) 𝜌 = 0.23, and (c) 𝜌 = 0.34. ................................................................................................................ 88 xiv Figure 5.2. (a) 3D printed H-ReH and R-ReH (𝜌 = 0.23). (b) Schematic of the experimental set- up of the drop tower tests. ............................................................................................................. 90 Figure 5.3. Standardized uniaxial tensile testing result of the printing material. ......................... 91 Figure 5.4. Typical stress-strain curves of H-ReHs and R-ReHs with different relative densities under (a) dynamic impact, and (b) quasi-static compression. ...................................................... 93 Figure 5.5. Comparison of dynamic and quasi-static stress-strain curves of H-ReHs. (a) 𝜌 = 0.23, (b) 𝜌 = 0.17, and (c) 𝜌 = 0.34. .................................................................................................. 96 Figure 5.6. The deformation patterns of H-ReH ( 𝜌 = 0.23 ). (a) Stress-strain curves, (b) deformation under quasi-static compression, and (c) deformation under dynamic impact. ......... 99 Figure 5.7. The deformation patterns of R-ReH ( 𝜌 = 0.23 ). (a) Stress-strain curves, (b) deformation under quasi-static compression, and (c) deformation under dynamic impact. ....... 101 Figure 5.8. SEA of H-ReHs and R-ReHs with different relative density under quasi-static and dynamic loading conditions. (Q is denoted for quasi-static condition, and D is denoted for dynamic condition). ................................................................................................................................... 102 Figure 6.1. Design and configurations of H-ReHs and R-ReHs with 𝜌 =0.34. (a) 45° H-ReH, (b) 45° R-ReH, (c) 60° H-ReH, and (d) 60° R-ReH. ........................................................................ 108 Figure 6.2. (a-b) 3D printed specimens with soft (black) and rigid (white) polymers: (a) Printed 45° H-ReH and 45° R-ReH with soft and rigid polymers. (b) Printed 60° H-ReH and 60° R-ReH with soft and rigid polymers. Constitutive behavior of the printing materials: (c) soft polymer (FLX4895), and (d) rigid polymer (Rigur). ................................................................................ 110 Figure 6.3. Setups for oblique compressive loading and axial loading. (a) Oblique loading with angle of 15º, and (b) axial loading with angle of 0º. ....................................................................111 Figure 6.4. Load-displacement curves of 45º ReHs under oblique (15º) and axial (0º) loadings. (a) ReHs fabricated by soft materials (b) ReHs fabricated by rigid materials. (O denoted for oblique loading, and A denoted for axial loading). .................................................................................. 113 Figure 6.5. Load-displacement curves of 60º ReHs under oblique (15ᵒ) and axial (0ᵒ) loadings. (a) ReHs fabricated by soft materials (b) ReHs fabricated by rigid materials. (O denoted for oblique loading, and A denoted for axial loading). .................................................................................. 114 Figure 6.6. Poisson’s ratio of H-ReHs and R-ReHs with different cell-wall angles and fabricated xv with different materials. (a) ReHs with 45º cell wall angle. (b) ReHs with 60º cell wall angle. ........................................................................................................................................... 116 Figure 6.7. Rotation analyses of 45º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. .......................................................................................................................................... 118 Figure 6.8. Rotation analyses of 45º R-ReH and H-ReH fabricated with rigid material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. .......................................................................................................................................... 118 Figure 6.9. DIC strain fields (major strain) of 45º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. ...................................................................................................................................... 120 Figure 6.10. DIC strain fields (major strain) of 45º R-ReH and H-ReH fabricated with rigid material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. ................................................................................................................ 121 Figure 6.11. Rotation analyses of 60º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 3.04 mm. (c) R-ReH and (d) H-ReH at displacement of 6.76 mm. ..................................................................................................................................... 123 Figure 6.12. DIC strain fields (major strain) of 60º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 3.04 mm. (c) R-ReH and (d) H-ReH at displacement of 6.76 mm. ................................................................................................................................. 125 Figure 6.13. Local deformation mechanism of 2nd order struts under the oblique loading. (a) Soft 45º H-ReH, and (b) soft 60º H-ReH. .......................................................................................... 125 Figure 6.14. Specific energy absorption of H-ReHs and R-ReHs under oblique and axial loading: (a) soft 45º ReHs, (b) rigid 45º ReHs, (c) soft 60º ReHs, and (d) rigid 60º ReHs. ..................... 127 Figure 7.1. (a) Design of multi-material H-ReH divided into outer and inner sections for different material assignment. (b) Stress-strain responses of multi-material H-ReH with hard, original, and soft outer section. ........................................................................................................................ 133 Figure 7.2. (a) Sandwich structure with H-ReH as new core material. (b) H-ReH filled tube as new thin-walled structure. .................................................................................................................. 134 Figure A-1. 15º Wedge design for oblique loading tests. ............................................................ 153 xvi Figure A-2. Long cap design of the 15º wedge for oblique loading tests. .................................. 154 Figure A-3. Short cap design of the 15º wedge for oblique loading tests. .................................. 155 Figure A-4. 30º Wedge design for oblique loading tests. ............................................................ 156 Figure A-5. Long cap design of the 30º wedge for oblique loading tests. .................................. 157 Figure A-6. Short cap design of the 30º wedge for oblique loading tests. .................................. 158 xvii CHAPTER 1. MOTIVATION To enhance the motility, lightweighting technologies are essential for the development of automotive and aerospace industries [1, 2]. As shown in Figure 1.1, the lightweight materials and structures have been widely applied in ground and air vehicle components, such as pillars, rails, chassis, and bumper systems. With these lightweight structures, the mass of vehicles is reduced, and thus the energy efficiency is significantly increased. It has been demonstrated that 10% mass reduction results in an improvement of fuel efficiency by 6-8% of ground vehicles [3] and 5-6% of aircrafts [4]. The development and application of lightweight materials and structures are an effective approach to reduce the fuel consumption and greenhouse gas emissions, which adversely contribute to the global warming, environmental pollutions and petroleum shortage [5, 6]. In addition, the lightweighting technologies endow vehicles with improved control, performance, and reduced material costs as well [7]. For the emerging vehicles with alternative energy sources, lightweight technologies are also desired, to improve the driving range. 1 Figure 1.1. Lightweight materials and structures used in ground and air vehicle body structures [8, 9]. Several types of conventional lightweight materials and structures have been manufactured and applied in automotive and aerospace industries and many other engineering practices, such as foams [10-14] and aerogels [15-18], as shown in Figure 1.2. The highly porous microstructures with interconnected network have provided foams and aerogels with unique properties such as low density, extremely low thermal conductivity, excellent mass transfer, etc [19, 20]. However, as shown in the Ashby plot in Figure 1.3, lightweighting leads to the reduction in both absolute and specific strengths of foams. This is due to the stochastic microstructure of foams, so that the material ligaments have an inferior load bearing capacity [21]. One important aspect of automotive design is the occupant’s safety under collision, which is directly determined by the strength and energy absorption capacity of the key components of the vehicle frames. To satisfy the safety needs, ideal lightweight materials exhibiting better mechanical properties at lower densities are desired. 2 Figure 1.2. Conventional lightweight materials with stochastic microstructures (a) Polyurethane foam [22]; (b) Microstructures of polyurethane foam [23]; (c) Aerogel monolith [24]; and (d) Syndiotactic polystyrene aerogel [25]. 3 Figure 1.3. Ashby map of strength versus density of the most common materials. (Reprinted from [26] with permission from John Wiley & Sons, Copyright 2018). Besides the matrix materials, material architectures play significant role on the development novel lightweight materials and structures. Natural architected materials with configured internal geometry exhibit exceptional high stiffness and strength as well as low density [27]. As shown in Figure 1.4, bones have efficient resistance to bending and buckling under various loading conditions and the wings of Morpho rhetenor butterfly have evolved layers of cuticle structure with great robustness for flapping. These superior mechanical properties are originated from the patterned microstructures with complex architectures. In addition, the beak of hornbill bird shown in Figure 1.4 (c) has a unique hierarchical structure, which benefits both the mobility and foraging of the hornbill bird. 4 Figure 1.4. Hierarchical materials found in nature. (a) Humerus bone structure of mice (Reprinted from [28] with permission from Oxford University Press, Copyright 2008); (b) Wing structure of Morpho rhetenor butterfly (Reprinted from [29] with permission from Springer Nature, Copyright 2003); (c) Beak structure of hornbill bird (Reprinted from [27] with permission from Annual Reviews, Copyright 2016). Inspired by nature, human have applied architecture design principle in large-scale structures to increase the structural efficiency [27]. As shown in Figure 1.5 (a) and (b), modern wheels constructed of equally spaced spokes have been used to replace primitive wheels at 3500 B.C. made of solid materials, which significantly enhances the mobility and reduce energy consumption of modern vehicles. Similarly in civil engineering, Eiffel Tower has employed advanced 5 lightweight structural design with great robustness to avoid the use of solid stone blocks for the construction of pyramid back at 2500 B.C (Figure 1.5 (c) and (d)). Figure 1.5. Architectures have been applied to structures to increase the structural efficiency. (a) Primitive wheel [30]; (b) Modern wheel [31]; (c) The pyramids of Giza [32]; (d) Eiffel Tower [33]. At smaller scale, similar structural design principle has been adopted to develop novel lightweight materials with superior mechanical efficiency. In the past few years, several lightweight materials with controllable cellular microstructures have been developed. As shown in Figure 1.3, the classical hexagonal honeycombs [34, 35] have much higher strength than the conventional foams. Due to the limitations of conventional fabrication methods, the initial design of architected materials only involves elementary microstructures. Advanced and complex 6 microstructures have the great potential to further enhance the mechanical properties as well as reduce the wight of architected materials. The recent rapid development in 3D printing technologies, also known as additive manufacturing methods, makes the fabrication of the advanced architected materials with more complex architectures possible. In this study, we aim to design a new type of lightweight architected materials with unprecedented mechanical properties with the aid of additive manufacturing. The specific architected materials focused of this dissertation is the hierarchical re-entrant honeycombs (H- ReHs), which combine the structural hierarchy and the advanced auxetic configurations. This dissertation focuses on the effects of geometric design, strain rate, and various loading conditions on the mechanical performance and deformation mechanism of the H-ReH. The findings presented herein provide thorough understandings of the structure-property relationship and facilitate future design of lightweight but robust architected materials. 7 CHAPTER 2. INTRODUCTION 2.1 Background 2.1.1 Architected Materials Architected materials, an emerging type of lightweight materials, have been developed based on the optimized configuration of solids and voids inside. With tunable and controllable architectures, the mechanical properties are manipulated to achieve ultralow density and high mechanical properties simultaneously. Honeycombs, one specific type of 2D architected materials with great easiness in manufacturing, have been widely used in various applications such as airplanes, vehicles, packaging, and sporting equipment [36], as shown in Figure 2.1. Honeycombs have a regular hexagonal-shape geometry (Figure 2.1 (a)), which not only minimizes the density of the material but also provide outstanding properties such as high energy absorption capacity, low thermal conductivity, excellent acoustic damping capability, and rapidness in manufacturing [34, 37, 38]. This is due to the controllable and tunable cellular structures, including the shape and size of the unit cell, and the aspect ratio and the angle of the cell walls [39, 40]. 8 Figure 2.1. Honeycombs and its various applications. (a) Honeycomb lattice structure. (b) 3D- printed honeycomb in a cross-section of the airplane wing. (c) Honeycomb applied in shoe insole. (d) Honeycomb structured automobile tire [41]. It has been demonstrated that the mechanical properties of lightweight materials are correlated with several factors, including (i) relative density, (ii) microstructure and geometry, and (iii) intrinsic material properties. Following the equation derived by Gibson and Ashby [42, 43], the mechanical properties of lightweight materials are predicted as: 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑔ℎ𝑡𝑤𝑒𝑖𝑔ℎ𝑡 𝑐𝑒𝑙𝑙𝑢𝑙𝑎𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝜌 𝑛 = 𝐶 (𝜌 ) (2.1) 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑜𝑙𝑖𝑑 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑠 where C and n are geometric constants, and ρ and ρs are the density of the lightweight materials and the bulk solid materials, respectively. It is noted that the value of n can vary from 1 to 4, 9 corresponds to different deformation modes and a smaller value of n represents a better performance of the material. As shown in Figure 2.2 (b), for lightweight materials with n = 1, all members are stretched under loading and the corresponding deformation mode is stretching- dominated. In contrast, for lightweight materials with higher values of n (Figure 2.2 (c)), the structural members undergo bending or buckling and the corresponding deformation mode is bending-dominated. Foams with random pores and stochastic structures typically has n ≈ 2, due to their inferior bending-dominated deformation mode [42]. Honeycombs with regular hexagonal geometry exhibit improved value of n (lower than 2) when density is relatively high. However, once reaching the lower density region, honeycombs have much increased value of n. This is due to the deformation mode shifts from yielding to cell wall buckling [27, 44]. To maintain the stretching-dominated deformation mechanism of honeycombs at lower density, advanced architecture design is needed. 10 Figure 2.2. (a) The scaling relationship between mechanical properties and density illustrating (b) stretching-dominated and (c) bending-dominated behavior exemplified by octet, honeycomb and metal foam. (Reprinted from [27] with permission from Annual Reviews, Copyright 2016). 2.1.2 Hierarchical Architected Materials Through a long adaptation process during the evolution, natural cellular materials, such as bones, tendons, wood, moss, and bamboo exhibit extremely high stiffness and strength and undergo stretching-dominated deformation [44-48]. This is because of the hierarchical topologies inside (Figure 2.3), which are achieved through refined spatial configuration of solids and voids inside the structure over multiple length scales. 11 Figure 2.3. Hierarchical structures of natural materials. (a) Bone (Reprinted from [49] with permission from Springer Nature, Copyright 2009); (b) Rattan wood (Reprinted from [50] with permission from Royal Society of Chemistry, Copyright 2009); (c) Moss (Sphagnum squarrosum) (Reprinted from [51] with permission from Royal Society, Copyright 2009); (d) Bamboo (Reprinted from [52] with permission from John Wiley & Sons, Copyright 2017); (e) Tendon [53]. Inspired by natural hierarchical materials, hierarchical architected materials have been extensively developed by replacing conventional cell walls of architected materials with structural hierarchy [54-58]. As shown in Figure 2.4, Meza et al. developed hierarchical octahedra lattice with much improved structural strength and stiffness [59]. In addition, they also created recoverable hierarchical octet lattice with ultralight and strong properties based on nanoscale ceramics [60]. Schaedler et al. fabricated ultralight metallic microlattice with structural hierarchy spans from nanometer to millimeter, which exhibits superior recoverability and energy absorption 12 capacity [61]. These research works have demonstrated effectiveness of structural hierarchy in mechanical enhancement of architected materials. However, as hierarchical architected materials have feature sizes ranging over multiple length scales, the fabrication and manufacturing of these hierarchical architected materials are extremely challenging and limited by high-end processing techniques, such as projection micro-stereolithography, atomic layer deposition (ALD), two- photon lithography (TPL), and oxygen plasma etching. Thanks to the rapid development of 3D printing techniques with improved printing resolution and capability, it is now possible to fabricate hierarchical architected materials with good quality, high efficiency, and less cost. 13 Figure 2.4. Examples of hierarchical architected materials simultaneously have ultralight, strong, and energy absorbing properties. (a) Hierarchical octahedra lattice (Reprinted from [59] with permission from National Academy of Sciences, Copyright 2015). (b) Ceramic octet-truss nanolattice (Reprinted from [60] with permission from The American Association for the Advancement of Science, Copyright 2014). (c) Metallic periodic hollow-tube microlattices (Reprinted from [61] with permission from The American Association for the Advancement of Science, Copyright 2011). 2.1.3 Auxetic Materials The origin of auxetic materials can be traced back to the 1980s, since when a variety of materials have been found to exhibit the unique negative Poisson’s ratio (NPR) effect, such as cubic metals [62], zeolites [63], foams [64], ceramics [65], and graphene [66]. This new type of materials contract transversally when undergoing compressive loading, and expand transversally under stretching, contrary to ordinary materials. As the auxetic behavior can potentially endow materials with enhanced mechanical properties, such as energy absorption capacity [67, 68], 14 indentation resistance [69], fracture toughness [70], and shear modulus [64], there are extensive interests in developing auxetic materials for various potential applications in textile [71-73], automotive [74], aerospace [75], and biomedical engineering [76, 77]. There are several advanced auxetic materials, including star-shaped [78] and V-shaped [79] honeycombs, chiral auxetic materials [80-83], origami and kirigami based materials [84-88], fractal cut metamaterials [89, 90], periodic unimode metamaterials of rigid bars and pivots [91], rotation of rigid hexamers, squares, and triangles [92-94]. Particularly, re-entrant honeycombs with inverted cells [95, 96] (Figure 2.5) have been developed and extensively studied because of their enhanced shear modulus [97, 98], indentation resistance [99], energy absorption [100], and electromagnetic properties [101]. However, the deformation mechanism of the auxetic re-entrant materials is still governed by bending-dominated behavior of ligaments, which results in suboptimal strength and stiffness of the materials compared with the linear scaling properties of bulk materials [21]. Figure 2.5. (a) 2D re-entrant honeycomb; (b) 3D re-entrant honeycomb lattice; (c) 3D re-entrant honeycomb unit cell (Reprinted from [96] with permission from Elsevier, Copyright 2015). 15 2.1.4 Combination of Hierarchical and Auxetic Structure As both structural hierarchy and auxetic behaviors are effective strategies for the design of lightweight materials and structures, hierarchical auxetic structures have the great potential to possess unprecedented mechanical properties. Until now, only a few studies have attempted to apply hierarchy into the auxetic re-entrant honeycombs, as shown in Figure 2.6. For example, vertex-based hierarchy [102, 103], star shape hierarchy [104], bidirectional hierarchy [105], hexagon hierarchy [106], and triangular hierarchy [106, 107] have been introduced into the re- entrant honeycombs. These studies have preliminarily demonstrated the enhanced performances of H-ReHs. However, a more comprehensive study with further understandings on structure- property relationship of H-ReHs is lacking. In addition, existing studies mostly reside in numerical simulations with very limited experimental validations [103, 106, 108]. This is mainly due to the challenges in fabrication process of H-ReHs. Nevertheless, thanks to the improved capability of advanced 3D printing techniques, H-ReHs are able to be fabricated by seamlessly integrating hierarchical and re-entrant honeycomb topologies together with desirable printing quality. As a result, it is now possible and viable to conduct a systematic investigation of H-ReHs from both experimental and numerical perspectives, aiming to provide thorough understanding of relations between hierarchical design, deformation mechanisms and mechanical responses, to fill the scientific gaps. 16 Figure 2.6. Hierarchical structures incorporated in the re-entrant honeycombs. (a) star-shape [104]. (b) Vertex-based (Reprinted from [102] with permission from Elsevier, Copyright 2021). (c) Bidirectional hierarchy (Reprinted from [105] with permission from Elsevier, Copyright 2022). (d) Hexagon and triangle hierarchy (Reprinted from [106] with permission from Elsevier, Copyright 2019). 2.1.5 3D Printing Techniques and Comparison The emerging 3D printing techniques, also known as additive manufacturing, is an efficient and scalable method to fabricate various structures and complex geometries with tunable architectures based on 3D models. The fabrication process of 3D printing involves deposition of printing materials in a successful layers manner. Due to the advantages of rapidness of prototyping, freedom of design, minimization of waste, and mass customization [109], 3D printing technique has been widely applied in industries of prototyping [110, 111], construction [112], and biomedical applications [113, 114]. A wide range of printing materials are available in 3D printing process, such as polymers, metals, ceramics, and concrete [115]. As the 3D printing method is able to 17 precisely engineer the microstructure of the materials, it can improve and even create new properties of the materials. For example, Chen et al. fabricated hierarchical honeycombs using commercial 3D printers with size features from micrometer to centimeter, which exhibit improved energy absorption capacity and shape integrity [116]. Peng et al. utilized an ink-based 3D printing method to fabricate ultralight biomimetic hierarchical graphene material, which exhibited exceptionally high stiffness and resilience [117]. As the proposed H-ReHs are composed of triangular hierarchy from micrometer level to centimeter level, the resolution and printing properties of the 3D printing techniques are critical to the mechanical performance of the resulted structures. It is necessary to compare different 3D printing techniques in details to select the appropriate 3D printing technique for the fabrication of H-ReHs. Among various printing materials available in 3D printing, polymers have advantages of less build time, cost, and post-processing [118]. In addition, there are many types of polymers available for 3D printing, such as polylactic acid (PLA), acrylonitrile butadiene styrene (ABS), polyethylene terephthalate (PET), and hybrid polymer materials, which provide customized material properties. Commonly used polymer 3D printing methods include fused deposition modelling (FDM), selective laser sintering (SLS), stereolithography (SLA), and material jetting (Polyjet). Each of them has specific advantages and disadvantages. For comparison, details of different polymer 3D printing characteristics are summarized in Table 2.1. Since the FDM and Polyjet 3D printing facilities are available at Michigan State University and used for prototyping in this study, these two methods are introduced and compared elaborately, as follows. 18 Table 2.1. A summary of specifications of polymer 3D printing methods. Polymer Type Material Resolution Printing Advantages Disadvantages References Form Order FDM Thermoplastic Filament 50-200 µm Point by 1. Inexpensive 1. Limited resolution [118-120] point 2. Multi-material printing 2. Poor surface quality SLS Thermoplastic Powder 80-250 µm Point by 1. High accuracy 1. Expensive [118-120] point 2. Fine details 2. Slow 3. Powder is recyclable SLA Thermoset Resin 10 µm Point by 1. Fast printing 1. High cost [118, 119] point 2. Fine resolution and 2. Slow high quality Material Thermoset Resin 16 µm Area by 1. Multi-material printing 1. Liquid form [121-123] Jetting area 2. High surface quality materials only 3. Fine resolution 2. Limited viscosity range 19 FDM is the most common polymer 3D printing method. In FDM method, thermoplastic polymer filament is used, which is heated at the nozzle of the printer to reach a flow state, and then deposited layer by layer on the platform (Figure 2.7 (a)). Afterwards, the fused filaments solidify by cooling down to room temperature. The FDM is compatible with various polymers, such as ABS and PLA. The main benefits of FDM include high speed, low cost, and simplicity of the process. However, FDM also has several drawbacks, including weak mechanical properties, poor resolution, poor surface quality of fused boundary (as shown Figure 2.7 (c-d)), and limited thermoplastic materials. Figure 2.7. FDM 3D printing technology and samples. (a) FDM system [124]. (b) FDM printer (Makerbot replicator 5th generation) in MSU library. (c) FDM printed hierarchical honeycomb using PLA. (d) FDM printed octet using ABS. 20 Material jetting (Polyjet) method is developed by combining the Inkjet technology and the use of photopolymers. Unlike the inkjet used in conventional printers by jetting onto papers, the liquid photopolymers ink is jet on the platform and then undergoes hardening when expose to UV lights. Therefore, material jetting method is achieved by jetting layers of curable liquid photopolymer onto a build tray firstly, and then instantly cured using UV lights (Figure 2.8). Moreover, multi- material and multi-color options are possible as the printer head has several nozzles, which is capable of simultaneously spraying various materials. When printing small-scale complex geometries, the printer also jets removable supporting material to ensure the printing quality. The supporting material is usually removed by water or solution bath. There are wide choices of materials for material jetting method, including rigid plastic, soft rubber like material, resins, and high temperature polymer materials. In comparison to the FDM method, the Polyjet method has several advantages, such as excellent printing resolution (up to 0.016 mm), good surface quality (no layer-by-layer appearance when using FDM method), and the capability of printing multi- material and multi-color in same parts. Therefore, Polyjet printing method is selected for this study, and all specimens are printed by Polyjet Objet Connex350 3D printer (Stratasys), as shown in Figure 2.8 (b). 21 Figure 2.8. Material jetting (Polyjet) printing technology and samples. (a) Schematic of material jetting (Polyjet) system [125]. (b) Polyjet printer (Objet Connex350) in Department of ECE in MSU. (c) Polyjet printed advanced Octet lattice and hierarchical re-entrant honeycomb using Rigur (thinnest wall is 0.5 mm) [126]. 2.1.6 Digital Image Correlation (DIC) The digital image correlation (DIC) is a powerful and practical non-contact optical technique for the full-field deformation measurement, which has been widely used in the field of experimental mechanics [127]. The DIC is based on digital image processing and numerical computation. The working of DIC involves comparing the digital images of a test sample at different stages of deformation [128]. To be more specific, firstly a square subset with pixels is selected from the reference image. Afterwards, the deformed subset is tracked from the deformed image through predefined criterion and optimization algorithm. In order for DIC to work 22 effectively, the square subset must have random and unique pixels with a range of contrast and intensity levels, which requires the test sample surface to be covered with natural or artificial speckle pattern. The speckle pattern creates random gray level intensity variation on the sample surface [129]. During the deformation of the sample, the speckle pattern deforms together with the sample surface, providing reliable surface deformation information. The images for DIC can be obtained from variety of sources at various length scale, such as consumer digital cameras, high- speed cameras, macroscopes and microscopes including scanning electron microscopes (SEM) and atomic force microscopes (AFM) [128]. An Aramis V8 3D-DIC system (shown in Figure 2.9) and GOM correlate professional software have been coupled with the mechanical testing of specimens in this study. This powerful 3D-DIC has high resolution of 4096 x 3000 pixels, frame rate of 25 fps (up to 100 fps), variable measuring area from 20 x 15 mm2 up to 5000 x 4000 mm2, and flexible dual LED lighting, which allows the DIC measurement on multiple length scales at various scenarios. 23 Figure 2.9. (a) Aramis V8 3D-DIC system; (b) DIC system coupled with universal test machine during mechanical testing. 24 2.2 Research Objectives The design of advanced lightweight structures relies on the fundamental understanding on the mechanical performance, and deformation mechanisms, which is still lacking. To fill this scientific gap, this dissertation aims to understand the relationship between geometric design, mechanical properties and the unique deformation mechanisms of H-ReHs under complex loading conditions. To this end, multiscale and multi-material H-ReH specimens have been fabricated through polymer-based high-resolution Polyjet 3D printing technique. The mechanical performance of the H-ReHs has been fully characterized by quasi-static compression tests, dynamic impact tests, and quasi-static oblique tests. To reveal the deformation mechanisms of the H-ReHs, high-speed camera and 3D-DIC have been utilized to analyze the internal strain field of the specimens under various loading conditions. The findings of this research shed light on the design of next generation ultralight but robust materials and structures. 25 2.3 Dissertation Outline The structure of the dissertation is as follows. Chapter 1 introduces the motivation of this research. Chapter 2 gives a comprehensive overview of existing lightweight materials and structures, including architected materials, hierarchical materials, and auxetic materials, and the detailed comparison of the current 3D printing techniques. Chapter 3 presents a strategic design of H-ReHs with 2nd order triangular hierarchy that exhibit superior mechanical properties under the quasi-static compression. The experimental results and the 3D-DIC analysis reveal the novel deformation mechanisms underpinning the superior mechanical properties of the H-ReHs. Chapter 4 elaborates the effect of cell-wall angle on the mechanical performance of H-ReHs. The research outcomes will guide the optimization of the geometric design and tunable mechanical performance of the H-ReHs. Chapter 5 investigates the crushing behavior of H-ReHs under the dynamic impact loading condition, which provides insights of the mechanical performance of the H-ReHs under the intermediate strain rates. Chapter 6 studies the mechanical behavior of H-ReHs under the quasi- static oblique loading conditions, so as to understand the governing deformation mechanism of H- ReHs under complex loading conditions. Chapter 7 summarizes the major findings of the dissertation and discusses the future directions of the research. 26 CHAPTER 3. 3D-PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB: ENHANCED MECHANICAL PROPERTIES AND THE UNDERLYING DEFORMATION MECHANISMS Re-entrant honeycombs are one type of lightweight cellular materials with superior energy absorption and impact resistance. Yet, the mechanical properties of re-entrant honeycombs need to be further improved due to its bending-dominated deformation mechanism. In this chapter, we design a novel stretching-dominated re-entrant honeycomb by combining the structural hierarchy and auxetic cellular configurations. Such hierarchical re-entrant honeycomb (H-ReH) is manufactured through fine-resolution 3D printing technique. The mechanical behavior and deformation mechanism of the designed H-ReH has been investigated through combined digital image correlation (DIC) and finite element simulation. The 3D-printed H-ReHs exhibit enhanced specific stiffness, specific initial-buckling strength, structural stability and specific energy absorption capacity due to the uniquely combined deformation mechanisms. A competition between the 1st order and the 2nd order hierarchy has been revealed, which governs the mechanical properties of the H-ReHs. These findings provide thorough understandings and facilitate future design of lightweight but robust cellular materials. 3.1 Introduction Over the past decades, enormous efforts have been invested in developing light-weight materials and structures with enhanced mechanical properties. These materials and structures can be applied not only as new core materials in sandwich panels [130, 131], thin-walled shell 27 structures [132, 133], and two-phase composites [134], but also has great potential for emerging applications such as soft robotics [135], soft electronics [136], shape morphing deployable materials [137], coronary stents [138], biomedical scaffolds [139], metamaterials [140], actuators and sensors [141], and microelectromechanical system (MEMS) [142, 143], so as to satisfy tremendous demand from rapid-advancing automotive [144], aerospace [145], defense [146], biomedical engineering [147] and energy related industries [148]. Specifically, honeycombs are of particular interests because of the combined lightweight nature, high energy absorption capacity, low thermal conductivity, excellent acoustic damping capability, and rapidness in manufacturing [27, 34, 38, 149, 150]. The superior mechanical performance of honeycombs in comparison to stochastic foams comes from the controllable cellular structures, including the shape and size of the unit cell [39, 40] and the aspect ratio and the angle of the cell walls [37]. In addition, the mechanical performance of honeycomb structures has been further improved by introducing auxetic cellular configurations, such as re-entrant, star-shape, and chiral unit cells [98-100, 105, 151-157]. However, the deformation mechanism of the auxetic honeycomb structures is still governed by bending-dominated behavior, similar to that of stochastic foams [42, 96]. It has been demonstrated that the bending-dominated cellular structures exhibit lower stiffness and strength than stretching-dominated ones, due to bending induced collapse of cell walls and lack of lateral constraints [21]. Inspired by natural cellular materials such as wood, bone and coral, hierarchical cellular structures have extremely high stiffness and strength and undergo stretching deformation [44-48, 59, 158]. The hierarchical topologies have been achieved through refining spatial configuration of 28 solids and voids inside the structure at multiple length scales. The manufacturing of hierarchical cellular structures is achieved but also limited by high-end processing techniques, such as projection micro-stereolithography, two-photon lithography (TPL), atomic layer deposition (ALD), and oxygen plasma etching. Nevertheless, the emerging additive manufacturing techniques enable the new design and efficient fabrication of the complex hierarchical cellular structures. By replacing conventional cell walls with structural hierarchy, hierarchical honeycombs have shown enhanced stiffness, energy dissipation and shape integrity, due to the stretching-dominated deformation mechanism [116, 159-166]. To this end, we introduce triangular hierarchy into the regular cell walls of re-entrant honeycombs. Previous studies mainly numerical simulations and analytical analyses have demonstrated the feasibility of incorporating hierarchical topology to improve the mechanical properties of the re-entrant honeycombs [102, 103, 106, 108, 167]. However, the underlying deformation mechanism and the dominated interaction between different order of hierarchies, are still elusive. In addition, experimental validation of the models is missing. In this study, through a combined experimental and numerical approach, we aim to improve the mechanical properties of the resulted structures by combining the advantages of structural hierarchy and the negative Poisson’s ratio effect, and to reveal the deformation mechanisms and hierarchical interactions governing the mechanical properties of the hierarchical re-entrant honeycombs (H-ReH). 29 3.2 Materials and Methods 3.2.1 Design of H-ReHs The configurations of both regular R- and H-ReH representative volume elements (RVE) are shown in Figure 3.1. The geometry of the R-ReH RVE was determined by three key geometric parameters, i.e. the width of the top horizontal member (W), the length of tilted members (L), and the member thickness (t), as shown in Figure 3.1 (a). The 2nd order structural hierarchy in the H- ReH RVE were specified by the thickness (tt) and the length (lt) of the triangular cell wall, as shown in Figure 3.1 (b). For all H-ReHs with different slenderness triangular hierarchy, the length of 2nd- order structural hierarchy (lh) is constant. For both R- and H-ReHs, the angle between horizontal and inclined members was set as 60ᵒ. To keep the volume of the RVEs a constant, two stacks of the 2nd order triangular units with same length were used in the H-ReHs to replace the solid struts in the R-ReHs. Figure 3.1. Design and configuration of (a) R- and (b) H-ReH representative volume elements. 3.2.2 Materials and Sample Preparation All ReH samples were manufactured using a Polyjet 3D printer (Object Connex350, Stratasys, 30 Ltd.). Rigur (RGD450, Stratasys, Ltd.), an engineered polypropylene-like material, was used as the printing material. To achieve best printing resolution and shape integrity, a gel-like support material (SUP705, Stratasys, Ltd.) was used. Upon completion of the printing process, the support material was carefully and completely removed by mild waterjet. As shown in Figure 3.2, typical ReH samples were composed of 2 × 3 RVEs with the overall size of 106 mm × 92 mm × 40 mm (width × height × thickness). The samples were thick enough to avoid out of plane buckling under quasi-static compressive tests. The details of the fabricated ReH samples were summarized in Table 3.1. The relative density of various ReH specimens, 𝜌̅ , was adjusted by the key geometric parameters by following Eqs. (3.1) and (3.2), respectively. (4√3𝐿+2√3𝑊−5𝑡)𝑡 𝜌𝑅−𝑅𝑒𝐻 = (2 (3.1) √3𝑊−√3𝐿−5𝑡)(√3𝐿+𝑡) 38√3𝑡 2 +69𝑙 𝑡 𝜌𝐻−𝑅𝑒𝐻 = 18(√3𝑙 2 +6𝑙 𝑡 𝑡 𝑡 𝑡 +3√3𝑡 2 ) (3.2) 𝑡 𝑡 𝑡 𝑡 For each relative density, at least 3 samples were prepared and tested. After printing, all samples were kept at room temperature for nine days to allow fully cure of the printing material. In addition, three standardized dog bone specimens (ASTM D638-14) and cylindrical specimens (ASTM D695-15) were printed and cured at the same conditions for the characterization of the mechanical properties of the printing material. 31 Table 3.1. The characteristics of the designed R- and H-ReH samples. 𝜌̅ W (mm) L (mm) t (mm) 𝜌̅ tt (mm) lt (mm) R-ReH 0.48 42.660 13.950 6.391 H-ReH 0.44 1.091 2.520 0.32 39.826 15.390 3.934 0.34 0.796 3.034 0.22 38.381 16.115 2.679 0.23 0.511 3.528 Figure 3.2. 3D-printed standardized specimen and ReH samples with different relative densities. (a) Standardized dog bone specimen and cylindrical specimen for uniaxial tensile and compressive tests, respectively. (b-d) R-ReH specimens (ρ̅ = 0.22, 0.32, and 0.48), and (e-g) H-ReH specimens (ρ̅ = 0.23, 0.34, and 0.44). The central RVE of each sample (highlighted) represents the DIC field of view. 3.2.3 Mechanical Testing Coupled with 3D-DIC The mechanical properties of the Rigur material and the 3D-printed ReHs were measured by a universal testing machine (Model 5982, Instron, Corp.) by uniaxial tensile tests (ASTM D638- 14) and quasi-static compressive tests (compression speed of 9 mm/min), respectively. The 32 Young’s modulus and initial-buckling strength of the printing material were determined from the measured stress-strain curve. For ReH samples, the obtained load-displacement curves were converted into stress-strain curves. The stress and strain were calculated as σ = F / A and ε = δ / h × 100 %, respectively, where F was the force applied by the Instron machine, A was the cross- sectional area of the sample, δ was the displacement of Instron machine crosshead, and h was the initial height of the sample. In order to evaluate the excellence of the ReHs, the stress was further normalized by the density of the sample as 𝜎𝑛 = 𝜎/𝜌 (3.3) The uniaxial compressive tests were coupled with a 3D-DIC instrument (Aramis V8, Trilion Quality Systems, LLC) to measure the strain field evolution. All samples were spray painted of 3- 7 pixel-sized black dots with 50% coverage to create patterns for DIC tracking. High-resolution deformation contours of the central RVE of each ReH sample was tracked and analyzed, as illustrated in Figure 3.2. Videos were captured at a rate of 1 fps and the strain fields and the deformation contours of ReH samples were computed by the GOM correlate professional software. 3.2.4 Finite Element Analysis Finite element (FE) analysis of the H- and R-ReHs under uniaxial compression were performed using Abaqus/CAE 2020. The material properties of the printing material are critical input for the FE models. As the properties are determined by the processing method, it is necessary to characterize the 3D-printed specimens under the same printing condition of the ReHs. By following the ASTM D638-14 and ASTM D695-15 standards, the constitutive tensile and 33 compressive behaviors of the printing material were measured and shown in Figure 3.3 (a) and (b), respectively. The initial linear portion of the stress-strain curves were used to determine the modulus, and the stresses at the end of the linear-elastic range were used to determine the strength. From the tensile test, the measured Young’s modulus ER and the tensile strength σR were 734.1 MPa and 43 MPa, respectively. In compression, the values were slighted changed to 749.3 MPa and 44 MPa. The density, ρR= 1200 kg/m3, and Poisson’s ratio, υR = 0.3, of the printing material are obtained directly from manufacture datasheet. Since the tensile and compressive behavior of the printing material was very similar, the tensile properties of the printing material were used in the FE models. The engineering stress-strain response of the printing material was converted into the true stress-strain one which was then imported into an elastic plastic isotropic model. For all R- and H-ReHs, 2 × 3 RVEs were applied for models to match the overall dimension of the 3D- printed samples. Both R- and H-ReHs were modelled using Abaqus/Standard package. For R- ReHs, the structure was finely meshed with ~110000 tetrahedral elements using 10-node tetrahedral solid element (C3D10). For H-ReHs, triangular cell walls of the 2nd hierarchy were meshed with ~50000 hexahedral elements using 8-node brick element (C3D8R). Convergence tests were conducted for R-ReHs and H-ReHs models to validate that the mesh size was appropriate (as shown in Figure 3.4), and the simulation results were accurate and reliable. In the FE models, the ReH was placed between two rigid plates. A general contact was defined for the boundary conditions between the ReH sample and the rigid plate, and the tangential friction coefficient was set as 0.2 [168]. The bottom plate was fixed, while the top one was movable in the vertical direction. During the compression, a uniform displacement was applied to the top plate 34 with a constant velocity (9 mm/min) to simulate the experimental testing condition. Both the top and bottom rigid plates were meshed with 1800 elements using 4-node bilinear quadrilateral element (R3D4). Figure 3.3. Standardized testing results of the printing material: (a) uniaxial tensile test and (b) uniaxial compressive test. Figure 3.4. Mesh sensitivity analyses for (a) R-ReHs and (b) H-ReHs. 35 3.3 Results and Discussion 3.3.1 Constitutive Behavior and Energy Absorption Performance of ReHs The mechanical properties of ReHs have been characterized by quasi-static compressive tests, and the resulting stress-strain curves are shown in Figure 3.5. In order to evaluate the excellence of the ReHs from the lightweight perspective, the stress was further normalized by the density (Figure 3.6). Figure 3.5. Quasi-static compressive response of R- and H-ReHs with different relative densities: stress-strain curves. As shown in Figure 3.6 (a-c), both R- and H-ReHs initially exhibit an elastic response. In general, the normalized stress response of ReHs is lower when the relative density is smaller. Two critical criteria for the lightweight design are the specific stiffness and specific initial-buckling strength of the structure. Therefore, the specific stiffness and specific initial-buckling strength are extracted and plotted separately in Figure 3.6 (d) and (e). For H-ReHs with 0.23 relative density, 36 the specific stiffness and specific initial-buckling strength are significantly higher than those of the corresponding R-ReH by 250% and 170% respectively. The enhancement is attributed to the better geometrical arrangements of structural members and the stretching-dominated deformation mechanism (detailed in the section 3.3.2). When the relative density of the ReHs is larger than 0.44, the advantages of structural hierarchy are diminished. This is because the member thickness of the R-ReH is getting close to that of the H-ReH. After reaching the initial-buckling strength, post-buckling stress plateaus are observed for both R- and H-ReHs. When the relative density is small, the H-ReHs have much higher specific post- buckling strength and larger fluctuations than the corresponding R-ReHs. The stress plateaus are terminated by either fracture or densification of the ReHs. The termination strain, 𝜀𝑡 , is defined as the strain at which the maximum energy absorption efficiency, 𝜂(𝜀), is reached [169, 170]. By following Eqs (3.4) and (3.5), the termination strain is calculated and plotted in Figure. 3.6 (f). 1 𝜀 𝜂(𝜀) = 𝜎(𝜀) ∫0 𝜎(𝜖) 𝑑𝜖 (3.4) 𝑑𝜂(𝜀) = 0, 𝑤ℎ𝑒𝑛 𝜀 = 𝜀𝑡 (3.5) 𝑑𝜀 37 Figure 3.6. Quasi-static compressive response of R- and H-ReHs with different relative densities. (a-c) Normalized stress-strain curves of R- and H-ReHs: (a) 𝜌̅ = 0.23, (b) 𝜌̅ = 0.34, and (c) 𝜌̅ = 0.44, (d) Specific stiffness, (e) Specific initial-buckling strength, and (f) Termination strain (D and F represent the termination mode of the ReHs is either densification or fracture). For R-ReHs, 𝜀𝑡 drastically decreases from 69% to 13% when the relative density increases from 0.22 to 0.48. In addition, only the R-ReH with 0.22 relative density is terminated in the densification mode, while the other two R-ReHs with larger relative densities are terminated in the fracture mode with early global failure. The ductile-to-brittle transition is triggered at a relative density in between 0.22 and 0.32. For H-ReHs, such transition takes place at a larger relative density in between 0.34 and 0.44. Moreover, when the H-ReHs are terminated in the densification mode, 𝜀𝑡 is independent from the relative density, as the H-ReHs with relative density of 0.23 and 0.34 have the same termination strain (~55%). To evaluate energy absorption capacity of ReHs, the specific energy absorption capacity (Es) 38 and the mean crushing force (Fm) are calculated by Eqs (3.6) and (3.7), respectively. ∫ 𝜎 𝑑𝜀 𝐸𝑠 (𝜀) = ̅ ∙𝜌𝑅 (3.6) 𝜌 𝐴∙∫ 𝜎 𝑑𝜀 𝐹𝑚 (𝜀) = (3.7) 𝜀 Figure 3.7. (a) Specific energy absorption and (b) Mean crushing force of R- and H-ReHs. As shown in Figure 3.7 (a), for similar relative densities, H-ReHs have significantly higher maximum specific energy absorption capacity, 𝐸𝑠𝑚𝑎𝑥 , than the corresponding R-ReHs. When 𝜌̅ ≈ 0.45, the 157% increase in 𝐸𝑠𝑚𝑎𝑥 is mainly due to the larger termination strain of the H-ReH. When 𝜌̅ ≈ 0.22, the 130% increase in 𝐸𝑠𝑚𝑎𝑥 is attributed to the much faster growth rate of Es. When 𝜌̅ ≈ 0.33, the increase in 𝐸𝑠𝑚𝑎𝑥 is further promoted to 250% as both 𝜀𝑡 and the growth rate of Es of the H-ReH are much larger than those of the R-ReH. The growth rate of Es is determined by the Fm. As demonstrated in Figure 3.7 (b), the Fm for all ReHs initially increases within the elastic region. During the post-buckling process, the Fm is nearly a constant for all ReHs. When the relative density is smaller, the difference between the Fm of the R- and H-ReHs is larger. 39 At smaller relative density, the stiffness of the 1st and the 2nd orders of hierarchy are comparable. As the two orders of hierarchy are interconnected with each other, the deformation of both hierarchies is competing with each other. This unique interaction between the hierarchies, which does not exist in R-ReHs, enables the H-ReHs to mitigate additional amount of energy. The interaction between the hierarchies facilitates the stable propagation of local deformation sites throughout the entire structure [171]. Therefore, the specific stiffness, mean crushing force, failure strain, as well as the specific energy absorption capacity of H-ReHs are significantly improved in comparison to those of R-ReHs. In addition, this interaction varies with the relative density that determines the difference between the stiffness of the hierarchies. When the difference is large enough, one hierarchy dominates the overall performance of the overall structure and the interaction between hierarchies diminishes. 3.3.2 Deformation Mechanisms of H-ReHs The improved mechanical properties of the H-ReHs are closely correlated to the unique deformation mechanisms associated with the hierarchical configuration. The negative Poisson’s ratio (NPR) of the ReHs is confirmed and calculated by tracking the movements of the joints under compressive tests, as shown in Figure 3.8 (a) and (b), respectively. As illustrated in Figure 3.8 (c), the νxy of both ReHs is negative and nearly constant in the elastic range deformation (ε ≤ 5%). However, at same relative density, the νxy of H-ReH is smaller than that of R-ReH. This is attributed to the different deformation mechanisms of the R- and H-ReHs. 40 Figure 3.8. Experimental images of (a) R-ReH (𝜌̅ = 0.32) and (b) H-ReH (𝜌̅ = 0.34) at ε = 0.05, and (c) Poisson’s ratio (νxy) for corresponding R-ReH and H-ReH. The strain fields of the central RVEs measured by the 3D-DIC are shown in Figure 3.9. For R-ReHs, all the horizontal and inclined members are under bending. Figure 3.9 (a) illustrates the strain field in the 0° direction. The top section of the horizontal member is in compression, while the bottom section is in tension. The central axis along the 0° direction of the horizontal member is nearly stress free as the neutral axis of a beam under bending. Figure 3.9 (b) and (c) display the strain field along the axis of inclined members (± 60°). Similar to the horizontal member, the inclined ones are in pure bending mode, as the inner and outer sections are under compression and tension, respectively. The compression and tension zones are separated by the neutral axis at the center of the members. This bending behavior is resulted from the rotation of the struts at joints [21, 172]. 41 Figure 3.9. DIC strain fields of R-ReH (𝜌̅ = 0.32) at ε = 0.05 (a-c) in 0°, 60° and – 60° directions, and H-ReH (𝜌̅ = 0.34) at ε = 0.04 (d-f) in 0°, 60° and – 60° directions. By contrast, both orders of hierarchy of H-ReHs exhibit stretching-dominated deformation. For the 2nd order of hierarchy, the triangular framework satisfies the topological criteria for stretching-dominated deformation mechanism [21], which is validated by the uniformly distributed strain fields in the struts of the 2nd order of hierarchy along the axis direction of the 1st order of hierarchy (Figure 3.9 (e) and (f)). All the struts are carrying loads axially without rotation at joints. As depicted in Figure 3.9 (d-f), the contours of all the 1st order of hierarchical members are nearly straight. All these members are composed of the stretching-dominated 2nd order triangular units. Besides, the members have enhanced bending stiffness, as the member weight is arranged further away from the neutral axis. The stretching-dominated deformation mechanism of the H-ReHs in the elastic range effectively improves the specific stiffness and initial-buckling strength than those of bending-dominated R-ReHs [108]. 42 Once the initial-buckling strength is reached, the ReHs undergo nonlinear deforamtion till the termination strain. However, the post-buckling behavior of R- and H-ReHs is governed by different deformation mechanisms. The typical deformation patterns of R- and H-ReHs are presented in Figure 3.10 (a-c). For R-ReHs, due to the bending-dominated elastic behavior, severe stress concentration is developed at the joints (Figure 3.10 (a)). When the applied compressive stress is higher than σi, early fractures (ε < 0.1) occur at vulnerable joints, as shown in Figure 3.10 (d). The fractured members neither carry the external load nor provide constraints to adjacent members, which leads reduced post-buckling strength of R-ReHs and compromised structural stability (Figure 3.10 (a-b)). With the increase of local fractures shear bands are fully developed (Figure 3.10 (b)), and thus the R-ReHs tend to fail in global fracture with a smaller termination strain. 43 Figure 3.10. Deformation patterns of (a) R-ReH (𝜌̅ = 0.32), (b) R-ReH (𝜌̅ = 0.48) and (c) H- ReH (𝜌̅ = 0.23). DIC strain fields of central RVEs of (d) R-ReH (𝜌̅ = 0.32) and (e) H-ReH (𝜌̅ = 0.23). By contrast, the post-buckling behavior of H-ReHs is dominated by the densification of the 2nd order of hierarchy instead of the fracture. As shown in Figure 3.10 (c), the stretching-dominated 44 triangular hierarchical struts of inclined members firstly exhibit local buckling behavior, followed by complete densification of the local 2nd order of hierarchy (Figure 3.10 (e)). As the local densification strain is small, the structural stability of H-ReHs in the nonlinear region is significantly enhanced. Consequently, the H-ReHs have higher post-buckling strength than R- ReHs. The local densification of the 2nd order of hierarchy is progressive from one site to another. With more and more densified 2nd hierarchy, stress concentration is built up around the joints and local fractures of the 1st hierarchy occur. The local densification of the 2nd hierarchy as well as the local fracture of the 1st hierarchy cause the fluctuation in the post-buckling stress plateau of H- ReHs. Please note that a shear band is not fully developed in the H-ReHs with small relative density as the global structural stability is enhanced and the local fracture of the 1st hierarchy is postponed by the local buckling of the 2nd hierarchy. Eventually, H-ReHs tend to be fully densified without global fracture at a larger termination strain. The combined higher specific stiffness and specific initial-buckling, higher post-buckling strength, and larger termination strain result in the superior energy absorption performance of the H-ReHs. It is noteworthy that the mechanical properties as well as the deformation mechanisms of H- ReHs depend on the slenderness of the 2nd triangular hierarchy, 𝑙ℎ /𝑡𝑡 . As shown in Figure 3.11 (a), the H-ReH (𝜌̅ = 0.23) has the slenderest triangular cell walls which tend to buckle at a lower stress. The additional spacing between the slender cell walls promotes the local densification of the 2nd hierarchy and the termination strain of the structure. The slenderness also reduces the bending stiffness and rigidity of the 1st hierarchy. Therefore, the 1st order members and joints are 45 more vulnerable to bending and rotation in the post-buckling region, leading to a relatively lower growth rate of specific energy absorption (Figure 3.11 (d)). On the other hand, the H-ReH (𝜌̅ = 0.44) with the lowest 𝑙ℎ /𝑡𝑡 has a much-enhanced resistance to local buckling, joint rotation and member bending. Therefore, this specific H-ReH exhibits the highest initial- and post-buckling strengths (Figure 3.6). However, the thicker triangular cell walls reduce the overall porosity of the structure. More importantly, as the 2nd hierarchy is stronger than the 1st one, local fracture at the joints instead of local densification of the 2nd hierarchy dominate the post-buckling behavior. Consequently, this H-ReH becomes “less hierarchical”, leading to compromised structural stability (Figure 3.11(c)), and smaller maximum specific energy absorption (Figure 3.11 (d)). Figure 3.11. Effect of the slenderness of the 2nd order triangular hierarchy on (a) – (c) the local deformation mechanisms at ε = 0.1, where 𝑙ℎ /𝑡𝑡 = 8.64, 5.54, and 4.04; and (d) specific energy absorption of the H-ReHs. Based on the above discussion, there is a trade-off between the buckling strength and the termination strain. H-ReHs with an optimal slenderness of the 2nd triangular hierarchy may reach 46 the balance between the buckling strength and the termination strain, and thus maximize the energy absorption and other mechanical performance. As suggested in Figure 3.6 and Figure 3.7 (a), the H-ReH (𝜌̅ = 0.34) with intermediate slenderness shows a better combination of strength and termination strain. The deformation mechanism is the combined local densification of the 2nd hierarchy and the fracture of the 1st hierarchy (Figure 3.11 (b)), which contributes to the highest maximum specific energy absorption and the largest termination strain, as shown in Figure 3.11 (d). Please note that in the current design strategy, lh is a constant and the slenderness of the 2nd order triangular hierarchy is simply determined by tt. Based on the fundamental understanding on the effect of slenderness on the structural performance, lh can be extended as an additional degree of freedom for future designs. The deformation patterns of the ReHs: (R-ReH (𝜌̅ = 0.22), H-ReH (𝜌̅ = 0.34) and (𝜌̅ = 0.44) are shown in Figure 3.12. DIC strain fields of H-ReHs and R-ReHs of all relative densities at ε = 0.04 in various directions are shown in Figure 3.13 – 3.16. 47 Figure 3.12. Deformation patterns of (a) R-ReH (𝜌̅ = 0.22), (b) H-ReH (𝜌̅ = 0.34) and (c) H- ReH (𝜌̅ = 0.44). 48 Figure 3.13. DIC strain fields of H-ReH (𝜌̅ = 0.23) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌̅ = 0.22) at ε = 0.04 (c) in 0°, and (d) 90°. 49 Figure 3.14. DIC strain fields of H-ReH (𝜌̅ = 0.34) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌̅ = 0.32) at ε = 0.04 (c) in 0°, and (d) 90°. 50 Figure 3.15. DIC strain fields of H-ReH (𝜌̅ = 0.44) at ε = 0.04 (a) in 0°, and (b) 90°, and R-ReH (𝜌̅ = 0.48) at ε = 0.04 (c) in 0°, and (d) 90°. 51 Figure 3.16. DIC strain fields of R-ReH (𝜌̅ = 0.32) at ε = 0.01 (a-c) in 0°, + 60° and – 60° directions, and H-ReH (𝜌̅ = 0.34) at ε = 0.01 (d-f) in 0°, + 60° and – 60° directions. 3.3.3 Numerical Simulation and Scaling Laws of ReHs Finite element method (FEM) is employed to extract the govening structure-property relationship of the ReHs. FE models for R- and H-ReHs are firstly generated and then validated by existing experimental results. The constitutive behavior of ReHs with relative densities in the range of 0.2 – 0.5 simulated by the FE models shows great agreement with the experimental data and is presented in Figure 3.17 (a). In addition, the deformation patterns of R- (bending dominated) 52 and H- (stretching dominated) ReHs within elastic region are also accurately captured by the models, as shown in Figure 3.17 (b) and (c). Figure 3.17. Numerical simulation of R- and H-ReHs. (a) Simulated constitutive behavior compared with experimental results, and simulated deformation patterns of (b) H-ReH, and (c) R- ReH. The simulated deformation patterns of H- and R-ReHs are shown in Figure 3.18. 53 Figure 3.18. Simulated deformation patterns of H- and R-ReHs. (a) H-ReH (𝜌̅ = 0.34), (b) H- ReH (𝜌̅ = 0.44), (c) R-ReH (𝜌̅ = 0.32), and (d) R-ReH (𝜌̅ = 0.48). It is known that the mechanical properties of cellular structures are determined by the relative density by following a scaling law [173]. To capture the scaling law for the ReHs, the validated FE models are used to predict the stiffness and initial-buckling strength of both R- and H-ReHs with wider relative densities range. The R- and H-ReHs with relative densities from 0.04 to 0.6 are developed by changing the thickness of the structural members only (as shown in Figure 3.1, t for R-ReH and tt for H-ReH). The relative stiffness and relative initial-buckling strength are ploted as functions of relative density by following the scaling laws in the format of 𝐸/𝐸𝑅 = 𝐴(𝜌/𝜌𝑅 )𝑚 and 54 𝜎𝑖 /𝜎𝑅 = 𝐵(𝜌/𝜌𝑅 )𝑛 , respectively, where A and B are geometry-dependent parameters, and m and n are scaling exponents. The scaling law parameters are presented in Table 3.2. All experimental and numerical data are ploted in log scale in Figure 3.19. It is observed that both relative stiffness and relative initial-buckling strength of all ReHs are follwing linear scaling relationships with the relative density. However, the scaling exponents for R- and H-ReHs are different. Specifically, the scaling exponents of relative stiffness of the R- and H-ReHs are 1.4 and 2.7, respectively (Figure 3.19 (a)). The mH is closer to 1, which indicates the stretching-dominated deformation mechanism of H-ReHs in the elastic range, and outperforms stretching-dominated alumina-polymer triangular micro lattice structure [174] and MAG octet-gyroid lattice structure [175]. The R-ReHs exhibit similar behavior to that of stochastic foams with mF = 3 [42], indicating the bending-dominated deformation mechanism. These results validate that the introduction of the 2nd order triangular hierarchy convert the bending-dominated R-ReHs into stretching-dominated H-ReHs. All the ReHs follow a similar scaling law for the relative initial-buckling strength versus relative density (Figure 3.19 (b)). The scaling exponents, n, for the R- and H-ReHs are very close, as both structures undergo local buckling when 𝜎 = 𝜎𝑖 . Please note that the superior post-buckling behavior of the H-ReHs over the R-ReHs cannot be captured by current model due to the complext deformation mechasnims. 55 Figure 3.19. Scaling laws for ReHs (a) Relative stiffness, and (b) Relative initial-buckling strength. Table 3.2. The scaling law parameters of relative stiffness and relative initial-buckling strength of ReHs versus relative density. Relative initial- Relative stiffness buckling strength A m B n R 1.21 2.7 1.18 2.7 H 0.73 1.4 1.17 2.4 3.4 Conclusions In the study, we have presented a strategic design of hierarchical re-entrant honeycombs that exhibit enhanced mechanical properties under quasi-static compression. Through a combined DIC analyses and numerical approach, novel deformation mechanisms governing the mechanical properties of hierarchical re-entrant honeycombs have been revealed. Major findings can be summarized as follows: (1) When the relative density is small, the H-ReHs have higher stiffness, initial-buckling strength, and specific energy absorption capacity than corresponding R-ReHs. When the 56 relative density is large, H-ReHs exhibit better structural stability, larger termination strain and specific energy absorption capacity than corresponding R-ReHs. (2) The superior properties of H-ReHs are endowed by the introduction of the 2nd order triangular hierarchy into ReHs. The additional hierarchy successfully converts the deformation mechanism from bending-dominated to stretching-dominated in the elastic range. In addition, a unique combination of local-buckling, densification and fracture deformation mechanisms of the H-ReHs is revealed, which contributes to significant improvement of the mechanical properties of H-ReHs. (3) Among the deformation mechanisms of H-ReHs, there is a competition between the 1st and the 2nd order of hierarchy, which is governed by the slenderness of the 2nd order triangular hierarchy. (4) The relative stiffness and the relative initial-buckling strength of both H- and R-ReHs follow scaling laws. The scaling exponents of the relative stiffness confirm the stretching- dominated behavior of H-ReHs and bending-dominated behavior of R-ReHs. Overall, this study provides the understanding of the structure-property relationship and the deformation mechanism of ReHs, which paves the way for the engineering design of next generation ultralight materials. 57 CHAPTER 4. ANGLE EFFECT ON THE MECHANICAL PROPERTIES OF 3D-PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB Hierarchical re-entrant honeycombs (H-ReHs) have demonstrated unique mechanical characteristics and superior energy absorption capacity. However, the optimized hierarchical design is yet to be explored so as to realize better mechanical properties of H-ReHs. In this chapter, we develop new H-ReHs by applying various angle designs on the 2nd order triangular hierarchy of H-ReHs. For references, regular re-entrant honeycombs (R-ReHs) with same angles without 2nd order hierarchy are characterized as well. The in-plane crushing behaviors of new H-ReHs are characterized under quasi-static compression tests. Results show that increasing the angle in H- ReHs can significantly enhance the elastic modulus, strength, structural stability, and energy absorption capacity. In addition, H-ReHs are much more sensitive toward angle designs compared to R-ReHs. By investigating the deformation mechanisms of H-ReHs, it is found that the interaction between the 1st order and 2nd order hierarchy is significantly affected by angle designs, which governs the enhancement of mechanical properties. These findings not only demonstrate great tunability of H-ReHs through angle designs, but also provide new understandings on the design of H-ReHs as lightweight safety devices. 4.1 Introduction The requirements of robust and lightweight materials have seen tremendous increase in the industries including aerospace [176], automotive [144], defense [146, 177], and biomedical sectors [147, 178]. Conventional cellular materials such as honeycombs have been widely used due to its 58 lightweight nature and decent mechanical properties [27, 179]. However, the in-plane crashworthiness of honeycombs is still weak [180]. In order to improve that, the auxetic re-entrant honeycombs (ReHs) have been proposed and extensively studied. With the unique deformation mechanisms, the ReHs exhibit excellent indentation resistance, fracture toughness and energy absorption capacity, which outperform the conventional honeycombs [70, 99, 100, 151]. It is noted that the mechanical characteristics of ReHs are not just determined by the base materials, but also dominated by the cell structure properties. Therefore, it is essential to explore the relationship between the cell structure design and crushing behavior of the ReHs so as to realize better architectural design of ReHs with enhanced properties for various applications [181]. There are several geometric parameters that govern the cell structure design of ReHs, such as aspect ratio, cell-wall angle, and volume fraction [182-184]. Among them, the cell-wall angle can be easily tuned to obtain new cell structures with distinct deformation modes and different auxetic properties. Recently, the cell-wall angle effect on the mechanical properties of the ReHs have been extensively researched. For example, Choudhry et al. found that cell-wall angles have significant effects on the compressive strength of the ReHs [155]. Hu et al indicated that the indentation resistance and negative Poisson’s ratio effect were enhanced with increasing the cell-wall angle [185]. Zhang et al. discussed the effect of cell-wall angle on plateau stress and energy-absorbed ability of ReHs under the dynamic crushing [182]. Wu et al. developed the graded cell-wall angle design of ReHs and achieved enhanced energy absorption capacity under low-velocity impact or quasi-static loadings [186]. Gu et al. studied the cell-wall angle effect on the tensile behaviors of the ReHs and revealed that ReHs with smaller angles had more significant auxetic property under 59 stretching [187]. These researches have demonstrated that the cell-wall angle have significant effect on the crushing behaviors and deformation mechanisms of the ReHs. Therefore, optimizing the cell-wall angle in the ReHs can effectively improve the mechanical properties of ReHs. In addition to the cell-wall angle, the hierarchical structures can improve the mechanical properties of ReHs as well. Natural hierarchical materials such as bone and wood, have exhibited amazingly high stiffness and strength [44, 45, 47, 188-190]. Their configurations are composed of structural hierarchy over multiple length scales. Inspired by this, hierarchical topologies have been also introduced into the ReHs as a novel design to improve their mechanical properties [107]. By substituting the solid ligaments of ReHs with the smaller, 2nd order hierarchical structures, the hierarchical re-entrant honeycombs (H-ReHs) have been developed, which exhibits a much- enhanced energy absorption capacity over regular ReHs [102, 104, 106, 107, 126, 167]. The enhancement of H-ReHs originates from unique local deformation of 2nd order hierarchy in coordination with the macro deformation of 1st order hierarchy. It is noteworthy that the geometric parameters of the H-ReHs can considerably affect the deformation mechanisms and the mechanical properties of the H-ReHs, and therefore is of great significance [102]. Since the cell-wall angle plays a prominent role on the deformation mechanisms and mechanical properties of regular ReHs, it is expected that tuning the angle design of hierarchical structures in H-ReHs has great potential to coordinate both local and macro deformation modes of H-ReHs, thus achieves distinctive mechanical properties. As the cell-wall angle can be easily adjusted, it is feasible to optimize the mechanical performance of the H-ReHs through fast prototyping. However, research exploring the effect of hierarchical angle design on the mechanical performance of H-ReHs, to the best of our 60 knowledge, is scarce and lack of experimental validations. In this study, we apply three different angle designs, namely 45º, 60º, and 75º, on the 2nd order triangular hierarchy in H-ReHs, then incorporate them into the 1st order hierarchy, the cell walls of the ReHs, thereby construct three hierarchical re-entrant honeycombs (H-ReHs) with different angle designs accordingly. For references, regular re-entrant honeycombs (R-ReHs) with same angles without 2nd order triangular hierarchy are characterized as well. Thanks to the advanced Polyjet 3D printing technique [191], the complex H-ReHs with various angle designs can be fabricated and experimentally characterized. Through quasi-static compression tests and strain field analyses, we aim to investigate the influences of angle designs on the mechanical properties and the deformation mechanisms of H-ReHs, reveal the underlying interaction between 1st and 2nd order hierarchy, thus ultimately shed light on the optimal angle design strategy for the H-ReHs. 4.2 Materials and Methods 4.2.1 Design of H-ReH and R-ReH with Different Angles Figure 4.1. shows configurations of 45°, 60°, and 75° H-ReH and R-ReH representative volume elements (RVE), respectively. For the design of H-ReH with different angles, the geometry of the H-ReH RVE was adjusted by tuning the angle between the horizontal and inclined struts of triangles as the 2nd order triangular hierarchy. After that, the angle between horizontal and inclined members of 1st order hierarchy was also adjusted accordingly to fit with 2nd order triangular hierarchy with the number of trianglular hierarchy remains constant. The dimension of the H-ReH 61 REV was determined by two geometric parameters, i.e. the width of RVE (𝑊𝐻 ), and the height of RVE (𝐻𝐻 ). The 2nd order structural hierarchy in the H-ReH RVE was specified by the thickness (𝑡𝑡 ) and the length (𝑙𝑡 ) of the triangular cell wall, as shown in Figure 4.1 (a). By contrast, the dimension of R-ReH RVE was determined by the width (𝑊𝑅 ), height (𝐻𝑅 ), and member thickness (𝑡𝑅 ). The out of plane thickness (T) of H-ReH and R-ReH samples are thick enough (15 mm) to avoid out of plane buckling under compressive tests. Please note that for straightforword comparison, all H-ReH and R-ReH are designed to have a same relative density of 0.34. Figure 4.1. Design and configuration of (a) 45°, (b) 60° and (c) 75° H-ReH and R-ReH (𝜌̅ =0.34) representative volume elements. 62 4.2.2 Materials and Sample Preparation All ReH samples were fabricated using a Polyjet 3D printer (Objet Connex350, Stratasys, Ltd.). A durable engineered photopolymers, Rigur (RGD450, Stratasys, Ltd), was used as the printing material. The constitutive behavior of the printing material is measured by following the ASTM D638-14 tensile test and shown in Figure 4.2 (a). The Young’s modulus is determined as 𝐸𝑅 = 734.1 MPa. The tensile strength is determined as 𝜎𝑅 = 43.0 MPa. The density of the printing material, 𝜌𝑅 , is 1200 kg/m3, and the Poisson’s ratio of the printing material is 0.3, which can be obtained from the manufacture datasheet. After jetting the printing material droplets onto the build platform, materials were cured by UV light to solodify. The temperature of printer extruder was kept at 70 °C during the printing process, while the printer platform was kept at room temperature. All samples were printed under same printing parameters. The typical H-ReH and R-ReH unit cell samples printed by Rigur material are shown in Figure 4.2 (b-d). All printed samples have a compatible overall dimension for 3D DIC analyses during the machenical testing. The geometric details of the fabricated H-ReH and R-ReH samples with different angles are summarized in Table 4.1. For each angle design, at least 3 samples were prepared and tested. After the finish of the printing, all samples were put in room temperature for 9 days to allow fully cure of the printing material before the testings. 63 Figure 4.2. (a) Constitutive behavior of the Rigur printing material. (b-d) 3D-printed H-ReH and R-ReH samples with different angles: (b) 45°, (c) 60°, and (d) 75°. Table 4.1. The characteristics of the designed H-ReH and R-ReH samples (𝜌̅ =0.34) with different angles. Angle W (mm) l (mm) T (mm) tt (mm) lt (mm) 45° 47.82 8.09 15 0.52 2.73 H-ReH 60° 47.63 11.51 15 0.72 2.73 75° 23.95 11.31 15 0.49 1.36 Angle W (mm) l (mm) T (mm) tR (mm) R-ReH 45° 47.64 9.10 15 3.04 60° 47.61 13.66 15 3.87 75° 23.83 12.57 15 2.20 4.2.3 Mechanical Testing Coupled with 3D-DIC The mechanical properties of the 3D printed H-ReH and R-ReH samples with various angle 64 designs were characterized by quasi-static compressive tests using a universal testing machine (Model 5982, Instron, Corp.) at a constant speed of 3 mm/min. From the measured load- displacement relationships, the stress-strain relationships can be obtained following the relationship: σ=F/A (4.1) ε = δ / h × 100 % (4.2) where F was the external force applied by the Instron machine, A was the cross-sectional area of the ReH samples, δ was the crosshead displacement of Instron machine, and h was the ReH sample height. A 3D digital image correlation (3D-DIC) measuring system (Aramis V8, Trilion) was used to measure the strain field evolution of samples during the mechanical tests. Before the DIC measurement, all samples were spray painted of 3 to 7 pixel diameter dots with 50% coverage using flat black general purpose spray paint (Rust-Oleum) for DIC tracking. The high-resolution video was tracked and recorded at a rate of 1 fps. Afterwards, the high-precision deformation contours of each ReH samples during quasi-static compression were computed by the ARAMIS Professional software. 4.3 Results and Discussion 4.3.1 Angle effect on the Elastic Beahvior of ReHs To characterize the mechanical properties of ReHs with different cell-well angle designs, quasi-static compressive tests have been conducted and the stress-strain responses are shown in 65 Figure 4.3. It is observed that initially both R-ReHs and H-ReHs with different cell-wall angles exhibit an elastic response, followed by a post-buckling behavior. Afterwards, while R-ReHs are terminated early by global fracture, all H-ReHs with different cell-wall angles are terminated in densification mode. Figure 4.3. Quasi-static compressive response of R- and H-ReHs with three different cell-wall angle designs: 75ᵒ, 60ᵒ and 45ᵒ. To investigate the initial elastic behavior of R-ReHs and H-ReHs, the negative Poisson’s ratio (NPR) effect of R- and H-ReHs with various cell-wall angles are investigated and shown in Figure 4.4. The Poisson’s ratios of ReHs are characterized by tracking the movement of the center tips. It is observed that increasing the cell-wall angles will increase NPR effect within elastic region for both R- and H-ReHs, as shown in Figure 4.4 (a). What’s more, when increasing the angle from 60 66 to 75, the increment of NPR effect is more dramatically. This is associated with the angle- dependent deformation mode change, which will be discussed in the following content. For all three cell-wall angles, the NPR effect of R-ReHs is always slightly higher than the H-ReHs. This is because R-ReHs are generally governed by a bending-dominated behavior and lateral displacement is more significant while H-ReHs are governed by a more stretching-dominated behavior [107]. 67 Figure 4.4. The negative Poisson’s ratio effect of R- and H-ReHs with different cell-wall angles at ε=0.05. (a) Poisson’s ratio (νxy) versus cell-wall angles. Experimental images of ReHs: (b) 45º, (c) 60º, and (d) 75º. In addition, the typical elastic modulus and strength of ReHs with different angles are shown in Figure 4.5 (a) and (b), respectively. For both H-ReHs and R-ReHs, it is evident that the elastic modulus and the strength are increased with increasing the angle. Such trends are logical as more structural components are aligned toward the vertical loading direction. However, H-ReHs exhibit a much greater angle-dependent behavior. To be more specific, H-ReH and R-ReH initially have 68 similar values of elastic modulus and strength at 45°, when increasing the angle to 60°, the elastic modulus and the strength of H-ReH are significantly enhanced by 120% and 90%, respectively, which is much higher than that of R-ReHs with only 38% and 4% enhancement, respectively. Such significant difference is attributed to the different interaction between the 1st order hierarchy and the 2nd order triangular hierarchy when increasing the angle of H-ReHs. As the strain fields of inclined members shown in Figure 4.5 (c), when increasing the angle, R-ReHs exhibit similar bending deformation mode on the inclined member. By contrast, the H-ReHs exhibit completely different deformation mode when increasing the angle. In 45° H-ReH, the 2nd order triangular hierarchy along the 45° axial direction shows stress concentration near the joints. In addition, the 1st order hierarchy (the inclined member) is not effectively carrying the compressive loading along the 45° axial direction, as the strain field only shows small area of compression. Therefore, the interaction between 1st and 2nd order hierarchy in 45° H-ReH is far from ideal. However, when increases the angle to 60°, the 2nd order hierarchy has a more continuous strain field distribution along the 60° axial direction (as highlighted) without stress concentration near the joints, and the 1st order hierarchy has increased area of compression. This demonstrates an enhanced interaction between 1st order and 2nd order hierarchy, and results in a significant enhancement on the elastic modulus and strength. When further increase the angle to 75°, further increased area of compression in 2nd order hierarchy of 75° H-ReH is observed, as highlighted. This is due to the increased NPR effect in 75° H-ReH (shown in Figure 4.4 (a)), thus leading to greater deformation of inclined member. In addition to the inclined axial direction, the hierarchical interaction along the horizontal 69 direction also exhibits enhancement. The strain fields of these H-ReHs along the horizontal direction are shown in Figure 4.5 (d). Due to the NPR effect, the tip joints tend to move inwards horizontally, and the 2nd order triangular hierarchy along horizontal direction is under stretching deformation naturally. Particularly, such stretching deformation exhibits an angle-dependent behavior as well. When the angle is 45°, the stretching is localized near the joints of horizontal struts and severe stress concentration can be observed. Once increasing the angle to 60° and 75°, the deformation is more uniformly distributed to the entire part of the horizontal struts. In general, both inclined and horizontal direction strain fields have demonstrated that increasing the angle can increase the interaction between 1st order and 2nd order hierarchy, which benefits the elastic modulus and strength comparing of the H-ReH. 70 Figure 4.5. Initial elastic behaviors of H-ReHs and R-ReHs with different angles. (a) Elastic modulus, and (b) Strength. DIC strain field in inclined members: (c) alone inclined member axial direction and (d) alone horizontal direction. For R-ReHs, the elastic collapse of the structure can be analyzed following the Euler– Bernoulli beam theory [155]. Figure 4.6 (a) shows the R-ReH structure under compressive stress 71 (σ) along y-direction. Due to the compressive stress, the inclined member of R-ReH exhibit a bending deformation, as shown in Figure 4.6 (b). The force applied in the inclined member (F) due to the compressive stress (σ) can be calculated as: 𝜎𝑤𝑇 𝐹= (4.3) 2 where w is the width of the horizontal member, T is the out-of-plane thickness of the R-ReH sample. Since the inclined member is in equilibrium, the force along y-direction is zero. Therefore, a moment (M) will be generated in the inclined member and can be calculated as: 𝐹𝑙 cos 𝜃 𝑀= (4.4) 2 where θ is the angle between horizontal and inclined members, l is the length of the inclined member. The deflection of the inclined member (δ) can be expressed by: 𝑀𝑙2 𝛿 = 6𝐸 (4.5) 𝑅𝐼 where 𝐸𝑅 is the Young’s modulus of the printing material, and I is the second moment of inertia 𝑇∙𝑡𝑅 3 of the inclined member and equals to , where 𝑡𝑅 is the thickness of the inclined member. 12 Afterwards, the strain of the R-ReH along y-direction can be calculated as: 2δcos 𝜃 𝜀= (4.6) 2𝑙 sin 𝜃 Since the elastic modulus (E) of R-ReH structure is: 𝜎 𝐸= (4.7) 𝜀 therefore, by substituting Eqs (4.3-4.6) into the elastic modulus Eqs (4.7) and rearranging, the final expression of elastic modulus (E) of R-ReH can be obtained as: 2𝐸𝑅∙ 𝑡𝑅 3 sin 𝜃 𝐸= (4.8) 𝑤𝑙2 cos2 𝜃 From this elastic modulus expression, it is evident that as θ increases, E will increase. Such 72 trend is consistent with the result shown in Figure 4.5 (a). Following the expression, the theoretical values of E are calculated and summarized in Table 4.2. Generally, the theoretical values of E are in good agreement to the experimental results, although 45° R-ReH has a slight discrepancy. The difference could be because part of vertical deformation applied by the compressive loading is carried by the horizontal member in 45° R-ReH, as 45° R-ReH has the longest horizontal member, which results in a relatively smaller vertical deformation and thus a greater value of experimental E. Nevertheless, the theoretical result here is a useful tool for the prediction of elastic modulus variation of R-ReHs when angle changes. Figure 4.6. (a) R-ReH structure under compressive loading, and (b) free body diagram of the inclined member in R-ReH under loading. 73 Table 4.2. Theoretical and experimental elastic modulus of R-ReH with different angles. Angle l (mm) w (mm) tR (mm) Theoretical E Experimental E (MPa) (MPa) 75° 12.57 15.71 2.20 90.82 93.66 60° 13.66 31.75 3.87 50.00 59.4 45° 9.10 24.84 3.04 28.4 43.07 4.3.2 Angle effect on the post-buckling beahvior of ReHs After the initial elastic region, it is found that the cell wall angle variations also have effects on the post-buckling behavior of R-ReHs and H-ReHs, respectively. As shown in Figure 4.7 (a), for R-ReHs, both 45° and 60° sample exhibit a stress plateau after the elastic region, while the 75° sample shows a sudden jump. The stress plateaus of 45° and 60° R-ReHs are associated with the bending and rotation of the inclined members in the structure, as shown in the stage 2 of Figure 4.7 (c), and stage b of Figure 4.7 (d), respectively. It is also observed that 45° sample has a slightly higher stress plateau with less stress drop over the 60° sample, this is because the 45° ReH has less rotation on the inclined members comparing with the 60° ReH. As highlighted in the red circles in Figure 4.7 (c), increasing the angle from 45° to 60° will increase the shear force around the joints between horizontal and inclined members, leading to the rotation of the red highlighted tips as well. On the other hand, unlike 45° and 60° samples, the 75° sample shows a sudden stress jump starting from stage Ⅱ on the stress-strain curves. This is associated with the contact of center tips of the 75° ReH (highlighted in Figure 4.7 (e) stage Ⅱ). Due to the NPR effect, the inclined members 74 tend to rotate and bend inwards under the compressive loading. Specifically, the 75° sample has the most significant rotation around the tips comparing to 45° and 60° samples, as highlighted in red circle in the Figure 4.7 (e), which results in the contact of the center tips. The contact provides additional support for the structure and leads to the stress jump. In general, the degree of bending and rotation on inclined members are increased when increase the angle of R-ReHs. After that, it is observed that the post-buckling behavior of R-ReHs are all terminated by the fracture of the inclined members, as shown in the last stage of Figure 4.7 (c-e), respectively. For 45° and 60° sample, they exhibit identical fracture mode with only one fracture point on each member, which is near the center tips of the inclined members due to the bending. Different from that, 75° sample have two fracture points on one inclined member, which leads to complete drop off of the inclined member and poor structural stability. To investigate the relationship between structural stability and cell-wall angles, the termination strain, 𝜀𝑡 , is calculated and plotted, as shown in Figure 4.7 (b). It is observed that 45° and 60° R-ReHs have a similar termination strain, while 75° sample have a reduced termination strain due to the severe bending. 75 Figure 4.7. Post-buckling behavior of R-ReHs with different angles. (a) stress-strain curves, (b) termination strain, and (c-e) deformation pattern of R-ReHs at various compressive stages: (c) 45°, (d) 60°, and (e) 75°. For H-ReHs, the post-buckling behavior are completely different from R-ReHs. As shown in Figure 4.8 (a), when the angle value of H-ReHs is relatively high (i.e. 60° and 75°), there are large fluctuations on the post-buckling stress plateaus, while for the lower angle of 45°, a lower stress plateau without fluctuations is observed. These are associated with the deformation mode change 76 when adjusting the angle of H-ReHs. As shown in Figure 4.8 (c), for 45° H-ReH, the 2nd order triangular hierarchy of four inclined members deformed simultaneously, exhibiting a collective deformation process: from the local fracture near the stress-concentrated joints (Figure 4.5 (c)), rotation, to the densification of 2nd order struts. However, since the length of inclined member in 45° H-ReH is short, there is barely the deformation of the 1st order hierarchy (the whole inclined member), and the densification of inclined members finishes rapidly (shown in stage 2 of Figure 4.8 (c)). Finally, once all of the 2nd order triangular hierarchy in inclined members are densified, 45° H-ReH reach its densification region (stage 3). By contrast, when increasing the angle to 60°, the H-ReH exhibits a localized, stepwise deformation mode. As shown in Figure 4.8 (d), initially at stage a, the 2nd order triangular hierarchy on the top two inclined members (highlighted) exhibits a progressive process of local buckling, fracture and densification behavior. In the meantime, the 1st order hierarchy, the inclined member as a whole, exhibit a first fracture and then contact with the horizontal members in the meantime. This is associated with first stress fluctuation around the position a on the stress-strain curve. Once contact, additional support helps the stress to go up again. Afterwards, the 2nd step of deformation mode is coming, as shown in stage b. The 2nd order triangular hierarchy on bottom two inclined members (highlighted) as well as the 1st order hierarchy itself exhibit a same local and global deformation behavior respectively. This is associated with second stress fluctuation on the position b of the stress-strain curve. As a result, the 60° H-ReH forms a symmetric pattern overally and the structural stability is retained. Please note this is different from 45º H-ReH where 1st order hierarchy barely deformed. Thanks to the effective deformation of the 1st order hierarchy, the 60° 77 H-ReH obtains a higher stress plateau from additional support and a extended stress plateau by avoiding global catastrophic failure of R-ReHs. Finally, the 3rd step of deformation mode is shown in stage c. It is observed that there are four shear band formed along the previously crashed 2nd order triangular hierarchy (highlighted), after which the rest of 2nd order triangular hierarchy starts to be crashed and densified. These deformation modes are responsible for the 3rd stress trough on the position c of the stress-strain curve. Lastly, 60° H-ReH reaches the densification region once all the 2nd order hierarchy is compacted. Similarly, 75° H-ReH also exhibits a stepwise deformation mode and a similar fluctuated stress-strain curves with 60° H-ReH. However, the fracture of 75° H-ReH exhibit an asymmetric pattern. As shown in Figure 4.8 (e), firstly the local buckling, fracture and densification of the 2nd order triangular hierarchy, together with fracture and contact of 1st order hierarchy, are observed on the top-left and bottom-right region of the inclined members, as shown in stage Ⅰ. Such deformation mode is assoctiated with the stress trough on the position Ⅰ of the stress-strain curve. Afterwards, the same process happen on the top-right and bottom-left region of inclined members, which is assoctiated with the stress trough on the position Ⅱ of the stress-strain curve. Different from 60° H-ReH, the asymmetric deformation pattern in 75° H-ReH is induced by the slight contact of center tips due to the NPR effect. However, thanks to the excellent structural stability, the 75° H-ReH shrinks transversely into a compacted rectangular shape, and exhibits an overally symmetric deformation pattern after the first two steps of deformation, as shown in the stage Ⅱ of Figure 4.8 (e). Following that, there are shear bands formed similar to 60° H-ReH (highlighted in stage III in Figure 4.8 (e)), which corresponds to the third stress trough at position Ⅲ on the stress- 78 strain curve. At last, the rest of uncrashed 2nd order triangular hierarchy undergoes compression and an overall densification is reached. It is noted that the degree of fluctuation for 75° H-ReH is slightly larger than the 60° H-ReH. This is because the geometrical components are aligned closer to the vertical loading direction, and the period between fracture to contact is longer in 75º H-ReH. It is observed that the stress plateaus of all H-ReHs are terminated by densification, instead of global fracture of R-ReHs. The termination strain, 𝜀𝑡 , is calculated at the strain with the maximum absorption efficiency of the structure by following Eqs (4.9) and (4.10): 1 𝜀 𝜂(𝜀) = 𝜎(𝜀) ∫0 𝜎(𝜀) 𝑑𝜀 (4.9) 𝑑𝜂(𝜀) = 0, 𝑤ℎ𝑒𝑛 𝜀 = 𝜀𝑡 (4.10) 𝑑𝜀 Different from R-ReHs, the 𝜀𝑡 of H-ReHs exhibit a clear angle-dependent behavior. As shown in Figure 4.8 (b), the 𝜀𝑡 increases from 40% to 65% when the angle increases from 45° to 75°. This is because there are more available interspaces between the 2nd order hierarchy along the loading direction when increasing the angle, which allows more vertical deformation, thus promotes the termination strain of the structure. This demonstrate that a brittle-to-ductile transition can be achieved by tuning the angle of the H-ReH. What’s more, the termination strains of H-ReHs are much greater than that of R-ReHs, i.e. when the cell-wall angle is 75°, there is 275% enhancement on the 𝜀𝑡 . This is due to the improved structural stability and the unique deformations of 2nd order triangular hierarchy. 79 Figure 4.8. Post-buckling behavior of H-ReHs with different angles. (a) stress-strain curves, (b) termination strain, and (c-e) deformation pattern of H-ReHs at various compressive stages: (c) 45°, (d) 60°, and (e) 75°. U means undeformed stage. 4.3.3 Angle Effect on Energy Absorption Capacity of ReHs To evaluate the angle effect on energy absorption capacity of H-ReHs, the specific energy absorption capacity (Es) is calculated by Eqs (4.11): ∫ 𝜎 𝑑𝜀 𝐸𝑠 (𝜀) = ̅ 𝜌𝑅 ∙𝜌 (4.11) 80 where 𝜌̅ is the relative density of the ReHs structure. As shown in Figure 4.9 (a), for R-ReHs, 45° and 60° sample has a very close growth rate of Es. When further increase the angle to 75°, the growth rate of Es is effectively enhanced. By contrast, for H-ReHs, the growth rate of Es is significantly enhanced when angle is increased from 45° to 60°. Further increase of the angle to 75° will lead to a very similar growth rate of Es with 60° H-ReH. It is also observed that with same angle, the H-ReH can always absorb more energy than R-ReH at all strain level. The maximum Es of the structure before its failure, 𝐸𝑠𝑚𝑎𝑥 , is calculated and shown in Figure 4.9 (b). For H-ReHs, when the cell-wall angle is higher (i.e. 60° and 75°), the 𝐸𝑠𝑚𝑎𝑥 can be significantly enhanced (290% increase) compared to the 45° H-ReH. This is due to the combination of larger termination strain and higher post-buckling stress of the H-ReH at the higher angle. It is also found that the 𝐸𝑠𝑚𝑎𝑥 of 60° and 75° H-ReHs is close. On the other hand, for R-ReHs, all three angles exhibit a similar 𝐸𝑠𝑚𝑎𝑥 , which means that changing angle will not have significant effect on the 𝐸𝑠𝑚𝑎𝑥 of R-ReHs. This is because of the trade-off between increment of stress levels and the reduction of termination strain of the R-ReHs when increases the angle. For higher cell-wall angle, it is noted that H-ReHs will have much higher enhancement on the 𝐸𝑠𝑚𝑎𝑥 compared with R-ReHs, as of 75°, the 𝐸𝑠𝑚𝑎𝑥 enhancement can reach 520%. This demonstrates great potential to improve the energy absorption capacity of H-ReHs by tuning the cell-wall angles. 81 Figure 4.9. (a) Specific energy absorption (Es) of R and H-ReHs with different angles. (b) Maximum specific energy absorption (𝐸𝑠𝑚𝑎𝑥 ) of R and H-ReHs as a function of the cell-wall angle. 4.4 Conclusions The mechanical proeprties of hierarchical re-entrant honeycombs (H-ReH)s with tunable cell- wall angle of 2nd order triangular hierarchy are yet to be explored. In this study, a series of H-ReHs and regular re-entrant honeycombs (R-ReHs) with three cell-wall angle designs and same relative density are fabricated and characterized. The effect of cell-wall angles on the compressive behavior of H-ReHs and R-ReHs under quasi-static compression tests have been investigated and compared. In addition, 3D-DIC analyses have been conducted to investigate the deformation mechanism of H-ReHs and R-ReHs with various angles. Finally, the specific energy absorption of ReHs with different angles have been compared. Major findings can be summerized as follows: • Within the elastic region, for both H-ReHs and R-ReHs, the elastic modulus and the strength are increased with increasing the angle. Interestingly, H-ReHs exhibit a much more angle- dependent behavior on the elastic modulus and the strength than R-ReHs. Such enhancement is found to be associated to the better combined deformation on both 1st order 82 and 2nd order hierarchy when increasing the angle. By contrast, R-ReHs have a similar bending deformation regardless of angle changes, and the elastic modulus of R-ReHs follows the prediction of standard beam theory. • In the post-buckling region, for R-ReHs, increase the angle will adversely result in an earlier termination strain. This is associated with the increased bending and rotation of inclined members and reduced structural stability when increase the angle of R-ReHs. By contrast, H-ReHs exhibit an beneficial angle-dependent post-buckling behavior. When increasing the angle, a transition from a low plateau into a higher fluctuated stress plateau is observed. This is associated with the enhanced effectiveness of both 1st order and 2nd order hierarchy. What’s more, increase the angle significantly enhances the termination strain of H-ReHs. It is noted that H-ReHs with various angles are all terminated by densification instead of fracture. • The maximum energy absorption capacity for H-ReHs is significantly increased with increasing the angle, while that for R-ReHs is almost innert to the angle change. Overall, cell-wall angle designs are found to have significant effects on H-ReHs due to the tunable and changeable deformation mode between different hierarchies, while R-ReHs are generally insensitive to angle variations and always dominated by a bending behavior. These findings demonstrate great tunability of H-ReHs through the cell-wall angle design, which shed light on the design of future lightweight hierarchical materials with favorable mechanical performance for targeted applications. 83 CHAPTER 5. STRAIN RATE EFFECT ON THE MECHANICAL PROPERTIES OF 3D-PRINTED HIERARCHICAL RE-ENTRANT HONEYCOMB Hierarchical re-entrant honeycombs (H-ReHs) with a 2nd order triangular hierarchy, an emerging type of lightweight yet robust materials, have exhibited prominent energy absorption performance under quasi-static compression. However, due to the structural strain rate effect as well as the viscoelastic nature of the printing material, the crushing behavior of H-ReHs under the dynamic loading condition could be completely different from the quasi-static scenario. Therefore, it is necessary to investigate the crushing behavior of H-ReHs under dynamic impacts, the real loading condition in crashworthiness applications. In this chapter, the dynamic responses of H- ReHs are characterized by a drop tower apparatus and compared with the quasi-static responses. For reference, regular re-entrant honeycombs (R-ReHs) are characterized as well. The results show that while R-ReHs exhibit similar post-buckling plateaus under different strain rate, H-ReHs exhibit a unique, two-step post-buckling plateau under dynamic impact, which is different from the quasi-static counterpart. What’s more, the linear elastic and the nonlinear post-buckling behaviors of the H-ReHs exhibit different levels of strain rate sensitivity. The strain rate-dependent behaviors are attributed to the micro-inertia effect and the localized deformation modes of the 2nd order triangular hierarchy. By contrast, R-ReHs without 2nd order hierarchy exhibit no deformation pattern change under different strain rates. Thanks to the combined material strain rate effect and the structural micro-inertia effect, the specific energy absorption capacity of H-ReHs under dynamic impacts can reach one orders of magnitude higher than that of the quasi-static one, and significantly outperforms the R-ReHs. These findings revealed the relationship between the 84 dynamic crushing behaviors of H-ReHs and the structural heterogeneity of the triangular hierarchy, which provides insights for the design of future lightweight but robust structures for crashworthiness applications. 5.1 Introduction During the past few decades, cellular materials such as foams and honeycombs have been proposed for energy absorption and shock mitigation applications, in order to satisfy the increasing demand on the impact protection in automotive, aerospace, and defense industries [27, 144-146, 192]. Since then, considerable researches have been conducted to investigate the mechanical responses of cellular materials under the quasi-static and dynamic loading conditions [42, 193, 194]. It has been demonstrated that the crushing of cellular materials under dynamic loading conditions is distinctive from the quasi-static one, mainly due to the dominated material strain rate effect and the structural inertia effect [195]. Under the dynamic loading condition, the stress required to crush the cellular materials is much higher than the quasi-static cases, and thus the energy absorption capacity of the cellular materials is significantly enhanced [196, 197]. In addition, the deformation patterns of the cellular materials are completely different under the dynamic loading condition, e.g. significant deformation localization than the quasi-static counterpart has been observed [198-200]. Therefore, it is necessary to investigate and compare both dynamic and quasi-static crushing behaviors of the cellular materials, and reveal the corresponding strain-rate dependent deformation mechanisms. Recently, auxetic re-entrant honeycombs (ReHs) with inverted cell walls have been 85 extensively investigated. Due to the unique negative Poisson’s ratio effect, ReHs have exhibited superior impact resistance over the conventional honeycombs [99, 153]. However, the mechanical properties of re-entrant honeycomb have not been optimized yet, due to its bending dominated behavior of the ligaments [96]. It has been demonstrated that the hierarchical structure topologies not only have great potential to convert the bending-dominated behavior of cellular materials into the stretching-dominated one, but also exhibit a combination of lightweight and excellent mechanical properties [47, 48, 59]. In light of this, the hierarchical structures have been introduced into the conventional cell walls, and the resulted hierarchical re-entrant honeycombs (H-ReHs) have demonstrated extraordinary mechanical properties [107, 126]. In the previous chapters, we have fabricated H-ReHs with triangular hierarchical substructures through 3D printing technique, and investigated their crushing behavior under the quasi-static compression. The 2nd order triangular hierarchy has a unique deformation behavior of elastic stretching, buckling, fracture and densification, which results in a much-enhanced energy absorption capacity over the regular re-entrant honeycombs. On the other hand, the crushing behavior of H-ReHs under the dynamic loading conditions, are yet to be investigated. As the dynamic loading conditions are the real loading conditions for crashworthiness applications, it is necessary to investigate the crushing behaviors of H-ReHs under the dynamic loading conditions. In this chapter, the mechanical responses of H-ReHs under the dynamic impact are investigated and compared with the quasi-static counterpart. It is found that while the stress levels are significantly enhanced for H-ReHs under the dynamic loading condition, the strain rate sensitivity of H-ReHs is different in linear elastic and the nonlinear post-buckling regions, respectively. In 86 addition, the energy absorption capacity for H-ReHs under the dynamic loading condition is significantly enhanced. These are associated with the micro-inertia effect and the deformation localization of the 2nd order triangular hierarchy in H-ReHs under the dynamic impact. 5.2 Materials and Methods The hierarchical re-entrant honeycomb (H-ReH) and regular re-entrant honeycomb (R-ReH) unit cells with three groups of relative densities: 0.17, 0.23 and 0.34 were designed and fabricated in this study, as shown in Figure 5.1 (a-c) respectively. The relative density was adjusted by changing the thickness of the hierarchical struts, namely tt of H-ReH and tR of R-ReH. The characteristics of H-ReH and R-ReH designs are shown in Table 5.1. The overall dimensions of all H-ReH and R-ReH specimens are the same: width (W) 60.4 mm, height (h) 35.5 mm, and out of plane thickness 18 mm. 87 Figure 5.1. Design of H-ReH and R-ReH with different relative densities. (a) 𝜌̅ = 0.17, (b) 𝜌̅ = 0.23, and (c) 𝜌̅ = 0.34. Table 5.1. Characteristics of designed H-ReH and R-ReH with different relative densities. 𝜌̅ tt (mm) 𝜌̅ tR (mm) 0.17 0.42 0.17 2.4 H-ReH 0.23 0.58 R-ReH 0.23 3.2 0.34 0.91 0.34 4.9 The H-ReH and R-ReH specimens were fabricated by the Polyject 3D printer (Objet Connex 350, Stratasys). The printing material was was mixture of Tangoblack and Verowhite (FLX9870- DM, Stratasys), which was a rubber-like polymer material. The typical printed H-ReH and R-ReH specimens are shown in Figure 5.2 (a). The constitutive behavior of the printing material was first characterized by uniaxial tensile test (ASTM D638-14). The typical constitutive behavior of the printing material is shown in Figure 5.3. The tensile strength was 5.0 MPa, and the elongation at 88 break was 164%. The mechanical properties of 3D printed H-ReHs were characterized by the quasi-static compression and the dynamic impact tests. For the quasi-static compression tests, a universal test machine (Model 5982, Instron) was used with a constant loading rate of 9 mm/min. The applied stress was calculated as σ = F / A, where F was the force applied by the Instron crosshead and A was the cross-sectional area of the 3D-printed H-ReH. The nominal strain was calculated as ε = δ / h × 100 %, where δ was the Instron machine crosshead displacement, and h was the initial height of the sample. Specific energy absorption (SEA) of H-ReHs was calculated ∫ 𝐹 𝑑𝛿 as 𝑆𝐸𝐴 = , where m was the mass of the H-ReH specimen. For the dynamic impact tests, a 𝑚 customized drop tower apparatus at the Ford Motor Company was used. Figure. 5.2 (b) shows the schematic of the drop tower setup. The incident speed of the dynamic tests was 2.5 m/s and the corresponding free drop height was set as 0.32 m. The drop weight was 8.8 kg. The deceleration time-history was collected by an accelerometer attached to the drop weight, and then recorded by an oscilloscope (PXIe-5105, National Instruments) at a sampling rate of 106 Hz. The stress applied to the H-ReH specimen was calculated as 𝜎 = 𝑚𝑑 ∙ 𝑎/𝐴, where md was the drop weight and a was the deceleration of the drop weight. The displacement was calculated as 𝛿 = ∬ 𝑎(𝑡) 𝑑 2 𝑡, where t was the time [201]. The displacement was then converted to the strain of H-ReH by using the sample height. During the dynamic impact tests, a high-speed camera (Chronos 1.4, Kron Technologies Inc.) was used to capture the local deformation of the H-ReHs at a frame rate of 5903 fps and a resolution of 640 × 360. To analyze the local deformation of H-ReHs under the dynamic impact, digital image correlation technique was employed. All 3D printed H-ReH specimens were firstly spray 89 painted with white 3-7 pixel size surface patterns (flat white, Rust-Oleum) for tracking [107]. After that, the high-speed videos recording the deformation process of H-ReHs during the dynamic impact were collected for the DIC analyses. By importing the high-speed videos into the DIC processing software (Aramis Professional 2019, GOM Company), the deformation contours of the H-ReHs under the dynamic impact were obtained and analyzed. Figure 5.2. (a) 3D printed H-ReH and R-ReH (𝜌̅ = 0.23). (b) Schematic of the experimental set- up of the drop tower tests. 90 Figure 5.3. Standardized uniaxial tensile testing result of the printing material. 5.3 Results and Discussion 5.3.1 Dynamic and Quasi-static Responses of H-ReHs The dynamic responses of H-ReHs and R-ReHs with different relative density are shown in Figure 5.4 (a). It is observed that both H-ReHs and R-ReHs exhibit an elastic response initially. The elastic modulus and the strength of H-ReHs and R-ReHs under the dynamic impact are summarized in Table 5.1. With all the relative densities, it is observed that both the elastic modulus and the strength of the H-ReHs are significantly higher than the R-ReHs. For example, when 𝜌̅ = 0.23, the strength of H-ReH can reach 5 times of the R-ReH under the dynamic impact conditions. This is associated with the unique deformation mode of H-ReH endowed by the 2nd order hierarchy, which outperforms the bending behavior of R-ReH [107]. After the elastic response of the structure, post-buckling plateaus are observed for both H- ReHs and R-ReHs. While for R-ReHs the post-buckling behavior is a one-step plateau with lower 91 stress level, the post-buckling behavior for H-ReHs exhibit a much higher, two-step plateau, which is associated with the localized progressive deformation of the 2nd order triangular hierarchy. The stress level of the post-buckling plateau is increased with increasing the relative density. Finally, all H-ReHs and R-ReHs exhibit a densification behavior. It is observed that the densification strain of R-ReHs is much lower than the H-ReH. For example, when 𝜌̅ = 0.34, the densification strain is 30% higher for H-ReH. This is associated with the unique local deformation of the 2nd order hierarchy in H-ReH, which promotes the overall deformation of H-ReH. For comparison, the typical stress-strain curves of H-ReHs and R-ReHs under quasi-static compression are shown in Figure 5.4 (b). Generally, all H-ReHs and R-ReHs exhibit a series of linear elastic, nonlinear post-buckling, and densification behaviors. For R-ReHs, the quasi-static responses are similar to that under the dynamic impact, exhibiting a flat post-buckling plateau. By contrast, the post-buckling stress plateaus of H-ReHs under quasi-static compression has significant difference compared with the dynamic impact one. It is observed that while the post- buckling behavior of H-ReH under dynamic impact has a two-step plateau in Figure 5.4 (a), in quasi-static scenario it switches to a one-step increment instead. Therefore, the dynamic impact effectively modify the mechanical responses of H-ReH. In addition, it is noted that the stress level of the dynamic responses is much higher than the quasi-static ones. As shown in Table 5.1 and Table 5.2, the dynamic elastic strength (σd) of H-ReHs are increased significantly by ~5000% compared with the quasi-static elastic strength (σq), with all relative densities from 0.17 to 0.34. Such increment is associated with the strain rate effect and deformation modes change of the H- ReHs under different loading rates. 92 Figure 5.4. Typical stress-strain curves of H-ReHs and R-ReHs with different relative densities under (a) dynamic impact, and (b) quasi-static compression. Table 5.2. Dynamic elastic modulus (Ed) and strength (σd) of H-ReHs and R-ReHs with various relative densities. Ed (MPa) σd (MPa) H 0.17 1.78 0.063 R 0.17 0.56 0.016 H 0.23 7.52 0.33 R 0.23 1.67 0.073 H 0.34 16.03 0.78 R 0.34 4.00 0.36 93 Table 5.3. Quasi-static elastic modulus (Eq) and strength (σq) of H-ReHs and R-ReHs with various relative densities. Eq (MPa) σq (MPa) H 0.17 0.067 0.0015 R 0.17 0.0030 0.0068 H 0.23 0.13 0.0053 R 0.23 0.015 0.0013 H 0.34 0.21 0.019 R 0.34 0.064 0.00032 In order to directly characterize the strain rate effect of H-ReHs with various relative densities, the dynamic and the quasi-static stress-strain curves for each relative density of H-ReH are plotted together, as shown in Figure 5.5 (a-c). To directly characterize the post-buckling behavior of H- ∫ 𝜎 𝑑𝜀 ReHs, the mean crushing stresses are calculated as 𝜎𝑚 (𝜀) = and plotted in Figure 5.5 as 𝜀 well. The dynamic stress range is 20 times of the quasi-static one for all these figures. As shown in Figure 5.5 (a), for the H-ReH with 𝜌̅ = 0.23, the dynamic stress-strain curve initially shows a much higher strength in the elastic region compared to the quasi-static curve. Afterwards, a stress drop on the beginning of the post-buckling region is observed. As the strain increases, the dynamic post-buckling plateau is getting closer with the quasi-static one, until reaching the final densification behavior. These results indicates that the linear elastic and the nonlinear post- buckling behaviors of the H-ReHs exhibit different levels of strain rate sensitivity. Similar 94 behaviors are also observed when the relative density of H-ReHs is 𝜌̅ = 0.17 and 𝜌̅ = 0.34, respectively, as shown in Figure 5.5 (b) and (c). It is observed that strain rate sensitivity in elastic region are always higher than the post-buckling ones. To further investigate the strain rate sensitivity of H-ReHs in elastic and post-buckling region, respectively, the elastic strain rate sensitivity (Se) and post-buckling strain rate sensitivity (Sp) are calculated, based on the ratios between the dynamic mechanical properties and the quasi-static ones. Firstly, the Se of H-ReHs are calculated as Eq (5.1): 𝑆𝑒=𝐸𝑑/𝐸𝑞 (5.1) where Ed is the dynamic elastic modulus, and Eq is the quasi-static elastic modulus of H-ReHs. Similarly, the Sp of H-ReHs are calculated as Ea (5.2): 𝑆𝑝=σ𝑚𝑑/σ𝑚𝑞 (5.2) where σ𝑚𝑑 is the dynamic mean stress at 40% strain, and σmq is the quasi-static mean stress 40% strain (Table 5.3). The resulted Se and Sp are summarized and shown in Table 5.4. It is observed that the value of Se is significantly increased with increasing relative density. This demonstrates that the elastic strain rate sensitivity of H-ReHs is strongly dependent on the relative density. By contrast, the Sp of H-ReHs almost remains constant (~30) with increasing relative density of the H-ReHs, as shown in Table 5.4. This, on the other hand, shows that the post-buckling strain rate sensitivity of H-ReHs is much less sensitive to the relative density. Furthermore, by comparing the values of Se and Sp within the same relative density of H-ReHs, it is found that the elastic strain rate sensitivity of H-ReHs is higher than the post-buckling one, which becomes increasingly significant when the relative density is higher. These are associated with the unique dynamic 95 deformation mechanisms of H-ReHs, which will be discussed in the next section. Figure 5.5. Comparison of dynamic and quasi-static stress-strain curves of H-ReHs. (a) 𝜌̅ = 0.23, (b) 𝜌̅ = 0.17, and (c) 𝜌̅ = 0.34. 96 Table 5.4. Dynamic mean crushing stress (σmd) and quasi-static mean crushing stress (σmq) at ε=40% of H-ReHs. σmd (MPa) σmq (MPa) H 0.17 0.045 0.0018 H 0.23 0.22 0.0061 H 0.34 0.71 0.023 Table 5.5. Elastic strain rate sensitivity (Se) and post-buckling strain rate sensitivity (Sp) of H- ReHs. Se Sp H 0.17 26.61 24.67 H 0.23 59.18 35.25 H 0.34 75.63 31.06 5.3.2 Dynamic Deformation Mechanisms of H-ReHs The deformation patterns of H-ReHs under quasi-static and dynamic loading conditions are compared and shown in Figure 5.6. The deformation patterns correspond to four points on the stress-strain curves shown in Figure 5.6 (a). It is observed that at small strain near the end of the elastic region (ε = 7.7%), the joints and struts of H-ReHs start to deform collectively under the quasi-static compression. By contrast, under the dynamic impact, most of joints and struts are 97 much less deformed at the same strain level. What’s more, localized joints and struts in the corner of H-ReHs are heavily deformed, as highlighted. This unique deformation mode is associated with the micro-inertia effect of the 2nd order triangular hierarchy under the dynamic impact, which resists the buckling and rotation of the local struts. Such micro-inertia effect suppresses the compliant buckling of struts under the quasi-static condition [202], thus not only leading to a higher yield strain of H-ReH under the dynamic impact (7.2%) comparing with the quasi-static one (4.9%), but also a much-enhanced strain rate sensitivity of H-ReHs in elastic region compared to the post-buckling one, as shown in Tables 5.3 and 5.4. When the strain further increases to ε = 20%, H-ReH under both quasi-static and dynamic impact exhibit a post-buckling plateau. This is associated with the struts in inclined members are heavily deformed, as shown in Figure 5.6 (b-c). However, the nearby struts in horizontal member (as highlighted) exhibits a much less compression under the dynamic impact. This is also associated with the micro-inertia effect inherited from the linear elastic region. As further increases the strain to 37.3%, the stress responses of H-ReH in both quasi-static and dynamic impact are increased. This is associated with the full compression of struts in inclined members, and the initiation of compression of struts highlighted in the central region. However, differences are observed between quasi-static and dynamic impact deformation patterns. While a compliant buckling mode of struts is observed under the quasi-static condition, an irregular buckling mode with counterrotation behaviors of certain struts is observed under the dynamic impact. These accounts for the unique two-step post-buckling behavior of H-ReHs under the dynamic impact, which is different from one-step post-buckling behavior under the quasi-static loading, as shown in Figure 5.6 (a). Lastly, both H-ReHs reach densification with local struts 98 compacted together when ε = 54%. Therefore, the strain rate has significant effect on the deformation modes of H-ReHs. Figure 5.6. The deformation patterns of H-ReH ( 𝜌̅ = 0.23 ). (a) Stress-strain curves, (b) deformation under quasi-static compression, and (c) deformation under dynamic impact. By contrast, for R-ReH the strain rate barely has effects on the deformation patterns. The deformation patterns of R-ReHs under quasi-static and dynamic loading conditions are compared and shown in Figure 5.7. It is observed that R-ReH has very similar deformation patterns when 99 increasing the strain rate from quasi-static one to dynamic impact one. Initially in elastic region, the R-ReH has center tips moving inwardly due to the NPR effect. As the compression proceeds, the joints connecting horizontal members and inclined members start to rotate, and inclined members exhibit an obvious bending behavior compatible with the further inward movement of center tips. Lastly, when ε = 50%, the inclined members of R-ReH completely contact with the horizontal members, exhibiting a densification behavior. In generall, the deformation patterns of R-ReH are independent with the strain rate. 100 Figure 5.7. The deformation patterns of R-ReH ( 𝜌̅ = 0.23 ). (a) Stress-strain curves, (b) deformation under quasi-static compression, and (c) deformation under dynamic impact. 5.3.3 Specific Energy Absorption of H-ReHs The specific energy absorption (SEA) of H-ReHs and R-ReHs under dynamic and quasi-static loading conditions are calculated and shown in Figure 5.8. The SEA values are obtained at the densification strain of each sample. Generally, with increasing the relative density, the SEA of both H-ReHs and R-ReHs are increased. Specifically, under the dynamic loading conditions, it is observed that the SEA of the H-ReHs are always higher than the R-ReHs with same relative density under the dynamic loading conditions, i.e. when 𝜌̅ = 0.23, H-ReH can have significant 350% 101 increment of SEA over R-ReH. This is due to the unique local deformation modes and micro- inertia effect endowed by the 2nd order hierarchy in H-ReH, which doesn’t exist in R-ReHs without the 2nd order hierarchy. In addition, under the quasi-static condition, the superior SEA of H-ReHs over R-ReHs are observed for all relative densities as well. These demonstrate that the 2nd order triangular hierarchy can effectively enhance the SEA of H-ReH in both dynamic and quasi-static loading conditions. What’s more, under the dynamic impact, it is found that H-ReH can absorb an order of magnitude higher mechanical energy over the quasi-static one. Such enhancement is attributed to the additional material strain rate effect together with the micro-inertia effect. Figure 5.8. SEA of H-ReHs and R-ReHs with different relative density under quasi-static and dynamic loading conditions. (Q is denoted for quasi-static condition, and D is denoted for dynamic condition). 5.4 Conclusion In this study, the mechanical performance of hierarchical re-entrant honeycombs (H-ReHs) 102 and R-ReHs under both dynamic and quasi-static loading conditions are investigated. Generally, the stress levels are significantly enhanced for both R-ReHs and H-ReHs under the dynamic loading conditions. However, H-ReH exhibit a unique strain rate-dependent post-buckling behavior compared with R-ReH. While R-ReHs exhibit similar one-step post-buckling plateaus under both loading conditions, H-ReHs under dynamic impacts evolves into a unique two-step post-buckling plateau, which is associated with the progressive localized deformation of the 2nd order triangular hierarchy. Furthermore, it is found that when the relative density is high, the H- ReHs exhibit a different strain rate sensitivity in linear elastic and the nonlinear post-buckling region respectively. Such difference is due to the micro-inertia effect of 2nd order triangular hierarchy in H-ReHs, which results in a localized deformation patterns of H-ReHs under the dynamic impact. By contrast, R-ReHs without 2nd order hierarchy exhibit no change on the deformation mode under different strain rates. With the combination of material strain rate effect and the structural micro-inertia effect, the H-ReHs under dynamic impact reach one orders of magnitude higher specific energy absorption (SEA) than the quasi-static one. In addition, the SEA of H-ReHs are significantly outperform the R-ReHs, confirming the effectiveness of the 2nd order triangular hierarchy under both dynamic impact and quasi-static loading conditions. These findings can shed light on the design of future lightweight but robust materials for crashworthiness applications. 103 CHAPTER 6. OBLIQUE LOADING EFFECT ON THE CRUSHING BEHAVIORS OF HIERARCHICAL RE-ENTRANT HONEYCOMBS MADE WITH SOFT AND RIGID POLYMERS Hierarchical re-entrant honeycombs (H-ReHs) with a stretching-dominated 2nd order hierarchy have demonstrated superior energy absorption capacity over regular re-entrant honeycombs (R-ReHs) under the axial loading conditions. However, in real crashing applications, the lightweight energy-absorbing materials not just experience a pure axial or transverse loading. Instead, an off-axial loading with a combination of axial and transverse loading cases, namely the oblique loading, is often the case. Different from the axial loading condition, the oblique loading conditions could strongly affect the mechanical responses of H-ReH. Therefore, it is important to understand the crushing behavior of H-ReHs under oblique loading condition. In this chapter, the quasi-static responses as well as the deformation mechanisms of H-ReHs and R-ReHs subjected to oblique loading have been experimentally investigated. The angle for oblique loading is 15º. The H-ReHs and R-ReHs with different cell-wall angles (45º and 60º) were characterized to investigate the effect of structural design on the oblique crushing behaviors. In addition, two printing materials with different rigidity, i.e. a soft polymer and a rigid polymer, were used to fabricate the H-ReHs and R-ReHs in order to explore the material effect on the crushing behavior under the oblique loading. The results show that the cell-wall angle design strongly affect the oblique crushing behaviors of H-ReHs. 45º H-ReH exhibit a unique enhanced mechanical response under oblique loading comparing with axial loading, while 60º H-ReH exhibit a reduced mechanical response under oblique loading, due to different deformation mechanisms. What’s 104 more, it is found that the oblique crushing behavior and deformation modes of H-ReHs strongly dependent on the rigidity of the constitutive material as well. When fabricated with rigid polymer, 45º H-ReHs has mild rotation with local shear band formation, which outperforms 45º R-ReH that has large rotation due to the plastic hinge formation. This demonstrates a hierarchical structure dominated behavior with rigid matrix material. By contrast, with soft polymer, 45º R-ReH reversely outperforms 45º H-ReH with less rotations due to softer and higher deformability of the matrix material. This, on the other hand, demonstrates hierarchical structure no longer dominates with soft matrix material. These findings provide valuable guidelines for design and material selection of lightweight energy-absorbing materials under complex loading conditions. 6.1 Introduction Over the past years, cellular materials such as foams and honeycombs have been widely used as energy absorber materials in automotive [144, 203], aerospace [145], packaging [204], and other industries, due to their lightweight nature and excellent energy absorption capacity during the crushing process [146, 205, 206]. The recently developed hierarchical re-entrant honeycombs (H- ReHs), an emerging type of cellular materials with the combination of hierarchical topologies and auxetic behaviors, have shown great potential to replace the conventional cellular materials [107]. It has been demonstrated that under the axial loading condition, H-ReHs exhibit much-enhanced mechanical properties, including specific stiffness, strength, energy absorption, and structural stability over the regular re-entrant honeycombs (R-ReHs), due to the unique deformation mechanism [102, 106, 107]. 105 For a well-designed energy-absorber material, it should be able to absorb maximum energy under various loading cases. Among which, the axial loading condition is only an ideal case. Actually, during a crash process in real complex engineering applications, the energy-absorber materials are not just undergoing net axial loading condition, but more likely subjected to the oblique loading condition at a loading angle, which is a combination of axial compression and shear loading [207]. For example, it is established that in automotive industry, the energy- absorbing bumper system should meet the requirement to be capable of bearing oblique load, as the non-ideal oblique loading condition is an unavoidable common situation [208, 209]. Therefore, it is critical to investigate the behavior of cellular materials under the oblique loading. It is known that the oblique loading can cause significant differences on the crushing behaviors of conventional honeycombs, including loss of energy-absorption capacity [210], and new deformation modes [211]. As for the H-ReHs, with the 2nd order hierarchy, both the geometric and constitutive properties should influence the oblique loading behaviors as well. However, existing studies mostly reside in the oblique behaviors of conventional honeycombs [211-214] and R-ReHs [215, 216], while the studies investigating the oblique behaviors of H-ReHs, to the best of our knowledge, is still limited. In this chapter, the quasi-static oblique responses of H-ReHs and R-ReHs are investigated at a loading angle of 15º and compared with the axial loading case. ReHs with different cell-wall angles design (45º and 60º) were characterized to investigate the effect of structural design. In addition, two printing materials with different constitutive behaviors were used to fabricate the ReHs to explore the material effect. The load-displacement performance and deformation 106 mechanisms of H-ReHs and R-ReHs are characterized and analyzed in detail. 6.2 Materials and Methods 6.2.1 Materials and Specimens Preparation The H-ReHs and R-ReHs with 45° and 60° cell-wall angle designs were investigated for the axial and oblique loading conditions. The design of H-ReHs and R-ReHs with different cell-wall angles are shown in Figure 6.1. The cell-wall angle was determined by the angle between horizontal and inclined structural components (θ). The overall dimension of H-ReHs and R-ReHs were determined by the width (WH and WR) the height (HH and HR), respectively. Specifically, the triangular hierarchy in H-ReHs were determined by two geometric parameters: length (lt) and thickness (tt), as shown in Figure 6.1 (a). While for R-ReHs, the member dimension was controlled by thickness (tR). For all H-ReHs and R-ReHs, the out of plane thickness was set as 15 mm to avoid out of plane buckling during the testing. All the geometric parameters are summarized in Table 6.1. The relative density of all specimens were maintained constant as 0.34. 107 Figure 6.1. Design and configurations of H-ReHs and R-ReHs with 𝜌̅ =0.34. (a) 45° H-ReH, (b) 45° R-ReH, (c) 60° H-ReH, and (d) 60° R-ReH. Table 6.1. The characteristics of designed H-ReHs and R-ReHs with different cell-wall angles. Angle W (mm) l (mm) T (mm) tt (mm) lt (mm) H-ReH 45° 47.82 8.09 15 0.52 2.73 60° 47.63 11.51 15 0.72 2.73 Angle W (mm) l (mm) T (mm) tR (mm) R-ReH 45° 47.64 9.10 15 3.04 60° 47.61 13.66 15 3.87 To fabricate the designed H-ReHs and R-ReHs, Polyjet 3D printing technique were used (Objet Connex350, Stratasys). Two photopolymers were used for the printing, one is black, soft 108 material with good deformability (FLX4895, Stratasys), the other is white, rigid material (Rigur, Stratasys). The specimens fabricated by soft and rigid materials are shown in Figure 6.2 (a-b). It is observed that the printed specimens exhibit good surface quality. The dimensions of specimens are compatible with the 3D DIC measurement during the mechanical testing. The constitutive behaviors of the soft and rigid materials were characterized by ASTM D638-14 uniaxial tensile tests, and results are shown in Figure 6.2 (c-d). For the soft material, the tensile strength is 10.5 MPa, and the elongation at break is 100%. For the rigid material, the Young’s modulus is 734.1 MPa, and the tensile strength is 43 MPa. The density of both soft and rigid materials is 1.2 g/cm3. During the printing process, the material droplets were jet onto the build tray, which were cured by UV light to solidify. For all specimens, the temperature of the printer extruder was set as 70 ºC, while the printer build tray was kept at room temperature. After printing, all specimens were put in the atmosphere condition with room temperature for 9 days to fully cure before the testing. 109 Figure 6.2. (a-b) 3D printed specimens with soft (black) and rigid (white) polymers: (a) Printed 45° H-ReH and 45° R-ReH with soft and rigid polymers. (b) Printed 60° H-ReH and 60° R-ReH with soft and rigid polymers. Constitutive behavior of the printing materials: (c) soft polymer (FLX4895), and (d) rigid polymer (Rigur). 6.2.2 Mechanical Testing and 3D-DIC Quasi-static oblique and axial compressive tests were conducted to characterize H-ReHs and R-ReHs using a universal testing machine (Model 5982, Instron). The experimental setups for oblique loading and axial loading are shown in Figure 6.3. Two wedges with 15º angle are installed and fixed onto both top and bottom Instron platens with bolting, as shown in Figure 6.3 (a). The top and bottom wedges are set to be parallel to each other before the testing. For axial loading, two horizontal platens were used without wedges, as shown in Figure 6.3 (b). The moving speed of top 110 platen was set as 3 mm/min for both oblique loading and axial loading. In order to fix specimens onto the wedges and platens and avoid sliding during the testing, a strong adhesive tape (F9460PC, 3M) was applied between the wedge surface and the specimen surface. For each ReH design, at least three samples were prepared and tested. During the quasi-static tests, a 3D digital image correlation (3D-DIC) system (Aramis V8, Trilion) was used to characterize the strain field evolution of specimens during the mechanical tests. Before the measurement, all specimens were speckled with 3 to 7 pixel size dots by using flat black/white spray paint with 50% coverage for DIC tracking purposes. During the measurement, the high-resolution deformation videos were recorded by the system at a rate of 1 fps, which will be analyzed and computed to generate high-precision deformation contours of specimens by using professional DIC software (GOM Correlate Pro, Zeiss). Figure 6.3. Setups for oblique compressive loading and axial loading. (a) Oblique loading with angle of 15º, and (b) axial loading with angle of 0º. 111 6.3 Results and Discussion 6.3.1 Constitutive Behavior of ReHs The mechanical responses of 45º ReHs fabricated with soft and rigid materials under oblique loading and axial loading have been characterized and results are shown in Figure 6.4. As shown in Figure 6.4 (a), when ReHs are fabricated with soft material, it is observed that under oblique loading, both H-ReHs and R-ReHs exhibit a series of elastic region, post-buckling plateau, and densification behaviors, similar to the axial loading curves. For R-ReHs, oblique loading reduced the force response of R-ReHs in elastic and post-buckling regions, which is similar to the existing oblique loading results in current literatures [211]. However, for H-ReHs, the oblique loading uncommonly increased the force response compared with the axial loading condition. This is because of different deformation patterns of H-ReHs and R-ReHs fabricated with soft material under the oblique loading, which will be discussed in the following section. In addition, it is noted that force response in elastic and post-buckling region of R-ReHs unexpectedly outperforms H- ReHs, demonstrating that 2nd order hierarchy no longer effectively improves the force responses of R-ReHs. At last, when reaching the densification behavior, the H-ReHs shows much higher displacement value over R-ReHs. This is because the effective local deformation of the 2nd order triangular hierarchy in H-ReHs, which promotes the overall deformability of the structure. By contrast, it is found that force responses of R-ReHs and H-ReHs are strongly dependent on the constitutive material. When the ReHs are fabricated by rigid material, distinctive force responses are observed. As shown in Figure 6.4 (b), the rigid H-ReHs under oblique loading exhibit enhanced force responses over the axial loading, while R-ReHs becomes weaker under the oblique 112 loading, similar to the soft ReHs cases. What’s more, superior force responses of H-ReHs are observed over R-ReHs under both oblique and axial loadings. This demonstrates that the 2nd order hierarchy fabricated by rigid material can effectively improve the force response of R-ReHs, which is different from the soft 2nd order hierarchy case. In addition, the R-ReHs fractured early and no longer exhibit a densification behavior, while H-ReHs exhibit a densification behavior. This is associated with unique local densification behavior from rigid 2nd triangular order hierarchy in H- ReHs, which is in agreement with our previous work [107]. The different mechanical responses of ReHs under oblique loading with material dependent behaviors are associated with the deformation mechanisms. Figure 6.4. Load-displacement curves of 45º ReHs under oblique (15º) and axial (0º) loadings. (a) ReHs fabricated by soft materials (b) ReHs fabricated by rigid materials. (O denoted for oblique loading, and A denoted for axial loading). The cell wall angle design also has effects on the oblique loading behavior of ReHs. The mechanical responses of 60º ReHs are shown in Figure 6.5. As shown in Figure 6.5 (a), for soft 113 material fabricated ReHs, it is observed 60º H-ReHs under oblique loading has inferior force response compared to axial loading, while 60º R-ReHs almost has identical force responses independent with loading angles. In addition, the H-ReHs always have higher force response over R-ReHs regardless of oblique loading or axial loading condition. On the other hand, when fabricated with rigid material, the post-buckling behavior changes. H-ReHs have fluctuations on the post-buckling plateaus followed by densification behavior, which is associated with the progressive local fracture and densification of 2nd order triangular hierarchy. By contrast, R-ReHs have global fractured and terminated at much lower displacement, due to the brittle nature of the rigid material. Therefore, post-buckling behavior of ReHs under oblique and axial loadings are correlated to the underlying deformation mechanism, which exhibit strong dependency with the constitutive material behavior. Figure 6.5. Load-displacement curves of 60º ReHs under oblique (15ᵒ) and axial (0ᵒ) loadings. (a) ReHs fabricated by soft materials (b) ReHs fabricated by rigid materials. (O denoted for oblique loading, and A denoted for axial loading). 114 6.3.2 Deformation Mechanism of ReHs under Oblique Loading To investigate the deformation mechanism of ReHs under the oblique loading, the negative Poisson’s ratio (NPR) effect of both H-ReHs and R-ReHs are characterized first. As shown in Figure 6.6 (a), for 45º ReHs, under the axial loading, R-ReH always exhibit a higher NPR effect than the H-ReH, which agrees with our previous research work [107]. By contrast, under the oblique loading, the NPR effect of R-ReHs are slightly increased compare to the axial ones, while H-ReH is almost insensitive to the loading direction. In addition, the soft R-ReH exhibit a higher NPR effect over the soft H-ReH, while the rigid R-ReH exhibit almost same NPR effect with rigid H-ReH. For 60º ReHs, the NPR effects are more significant than 45º ReHs. Under the axial loading, the R-ReH has an increasingly lower NPR value than the corresponding H-ReH, for both soft and rigid ReHs. By contrast, under the oblique loading, it is noted that while soft ReHs remain the same NPR values with the axial ones, the Rigid ReHs exhibit much reduced NPR values, which is associated with the serious rotation effect of ReHs under the oblique loading. The clear loading direction-dependent behavior and material-dependent behavior of ReHs under the oblique loading are associated with the different deformation mechanisms and will be discussed in the following contents. 115 Figure 6.6. Poisson’s ratio of H-ReHs and R-ReHs with different cell-wall angles and fabricated with different materials. (a) ReHs with 45º cell wall angle. (b) ReHs with 60º cell wall angle. It is found that under oblique loading, the ReHs not only has NPR effect, but also has a rotation effect. This is because during the oblique loading, the position of both bottom surface and top surface of the specimen are fixed with the loading wedges, thus with the proceeds of the oblique compression, the off-axis loading will generate additional shear force on the specimen, leading to a rotation effect of the sample, as the left center tips moves upwards, while the right center tips moves downwards, as shown in Figure 6.7. In order to investigate the rotation effect of ReHs under the oblique loading condition, the rotation effects of ReHs are evaluated by characterizing the center tip distance along the y1 direction of the specimen at certain compressive displacement. The rotation effects of ReHs fabricated with soft materials are shown in Figure 6.7. It is observed that at same overall compressive displacement, the R-ReH has a slightly smaller tip distance value compared with the H-ReH. This means that the rotation of soft R-ReH is slightly smaller than the H-ReH, 116 demonstrating that the soft 2nd order hierarchy doesn’t play significant role on the rotation of the structure under the oblique loading. Thus, for soft ReHs the material effect is dominated, regardless of the structural design. However, when the ReHs are fabricated with rigid material, the ReHs exhibit completely different rotation effects. As shown in Figure 6.8, the R-ReH fabricated with rigid material exhibit much larger tip distance value, which is 3 times of the soft R-ReH shown in Figure 6.7. This is associated with the seriously increased counter rotation of center tips together with the connected inclined members of R-ReH. This demonstrates that switch the material from soft to rigid will significantly increase the rotation on the R-ReH. By contrast, for rigid H-ReH, the tip distance is much smaller than the corresponding rigid R-ReH, as shown in Figure 6.8 (b) and (d). The rotation effect is significantly decreased by 56% for rigid H-ReH compared with the corresponding R-ReH. This, on the other hand, demonstrates that the rigid 2nd order triangular hierarchy can effectively reduce the rotation of H-ReH and promote the structural stability under the oblique loading. In such case, the structural effect is dominated for the rigid ReHs. Therefore, the crushing behavior of ReHs under oblique loading are strongly dependent on both the constitutive behavior of the material and the structural effect. 117 Figure 6.7. Rotation analyses of 45º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. Figure 6.8. Rotation analyses of 45º R-ReH and H-ReH fabricated with rigid material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. The combined NPR and rotation effects would result in significant difference on the deformation mechanisms of ReHs under the oblique loading. In order to characterize that, the 118 strain field of H-ReH and R-ReH are characterized by DIC techniques. The strain field of soft ReH are shown in Figure 6.9. For R-ReH, it is clearly shown that the NPR effect leads to severe bending deformation on the inclined member, as one side are heavily stretched while across the neutral axis the other side are heavily compressed, as shown in Figure 6.9 (a). As further compressed, the rotation effect under the oblique loading becomes increasingly notable, which intensify the stretch of the inclined members. In comparison to R-ReH, the soft H-ReH appear to exhibit a different deformation pattern. As shown in Figure 6.9 (b), initially when the compressive displacement is low, the deformation are concentrates and localized on the local joints connecting the 2nd order struts in the inclined members. When further compressed, the compression of 2nd order struts is exacerbated, and the local struts are severely buckled. In addition, with the increased rotation effect, the 2nd order struts located in top-right and bottom-left corner of H-ReH are deformed heavily as well. As the soft material enhance the stretching of the inclined members in R-ReH, while diminished the 2nd order struts with an inferior buckling deformation in H-ReH, the R-ReH surprisingly exhibit a higher force response over the soft H-ReH (Figure 6.4 (a)), contradicting with the conventional advantages of the hierarchical structures. These deformation behaviors account for the higher NPR effect and lower rotation effect of R-ReH compared with H-ReH. 119 Figure 6.9. DIC strain fields (major strain) of 45º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. On the other hand, the ReHs fabricated with rigid materials exhibit completely different deformation patterns under oblique loading. As shown in Figure 6.10 (a), for R-ReH, the deformation are localized in bottom-left and top-right inclined members, forming two plastic hinges rotating in counter-direction, leading to a much larger rotation compared with the soft R- ReH in Figure 6.9 (a). Upon further oblique compression, the plastic hinges deform and rotate more severely, followed by horizontal member fully contacting with the top-left and bottom-right inclined members, respectively. By contrast, the rigid H-ReH exhibit a different deformation pattern with much reduced rotation effect. As shown in Figure 6.10 (b), the 2nd order struts avoid the formation of plastic hinges on the inclined members. Instead, the 2nd order struts within the top-left and bottom-right members exhibit severe localized stretching deformation. When the oblique compression proceeds further, there are local fracture of 2nd order struts, and two shear 120 bands are formed in top-left and bottom-right part of the inclined member. Such deformation modes not only provide extra support to the H-ReH, but also effectively enhance the rotation resistance of the structure. Therefore, the force response of rigid H-ReH is superior to the rigid R- ReH (Figure 6.4 (b)). In general, the soft material restrains and eliminates the effectiveness of the 2nd order hierarchy in 45º H-ReH, leading to a material-dominated behavior of 45º ReHs. To the contrary, the rigid material promotes the effectiveness of the 2nd order hierarchy, leading to a structural-dominated behavior of 45º ReHs. Figure 6.10. DIC strain fields (major strain) of 45º R-ReH and H-ReH fabricated with rigid material. (a) R-ReH and (b) H-ReH at displacement of 2.34 mm. (c) R-ReH and (d) H-ReH at displacement of 4 mm. It is found that the above soft material-dominated effect is also dependent on the cell-wall angle design of ReHs. In order to characterized it, the ReHs with 60º cell-wall angles are characterized under the oblique loading as well. The rotation analyses of 60º ReHs are shown in 121 Figure 6.11. Initially at relatively small displacement, R-ReH exhibit a clear inward deformation with inclined members undergo bending deformation due to the NPR effect, as shown in Figure 6.11 (c). In the meantime, the center tip distance is small due to the rotation effect. Once the displacement is increased, inclined members are further bended and rotated, similar to the rotation behavior of soft 45º R-ReH (Figure 6.7 (c)). On the other hand, for 60º H-ReH, the local joints connecting inclined member and horizontal member start to twisted, leading to the connected 2nd order struts rotate in a compliant mode. With increased displacement, the local joints severely twisted, and 2nd order struts rotate heavily to densified with each other. With the joint twisting and struts rotation deformation mode, H-ReH exhibit slightly higher rotation effect over the R-ReH. However, the soft 60º H-ReH exhibit better force response over the corresponding R-ReH, as shown in Figure 6.5 (a). This means the soft material is not always restrain the effectiveness of 2nd order hierarchy, and the hierarchy angle design plays a significant role here. 122 Figure 6.11. Rotation analyses of 60º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 3.04 mm. (c) R-ReH and (d) H-ReH at displacement of 6.76 mm. To further understand that, the strain field of soft 60º ReHs are characterized. As shown in Figure 6.12 (a), initially the inclined members in R-ReH have a bending deformation mode, with small region adjacent to the center tips stretched. As the compression increased, the bending deformation becomes increasingly notable, and much larger region of the inclined member are heavily stretched. This is similar with the 45º R-ReH as previous shown Figure 6.9 (c). As for H- ReH, the local joints connecting inclined member and horizontal member exhibit clear twisting, as shown in Figure 6.12 (b). In addition, deformation localizations are observed at the tips of the connected struts, due to the compliant rotation deformation mode of the struts. Upon further 123 compression, these struts are extensively rotated and stretched, so that they self-compacted to densified with each other. It is noted that the localized behavior of 2nd order struts in 60º H-ReH is distinctive with the soft 45º H-ReH. As shown in Figure 6.13 (a), in soft 45º H-ReH, under the oblique loading, there is a combination of compression and rotation effect. Due to the softness of the material and smaller angle between inclined and horizontal struts, the 2nd order struts are more vulnerable to the compression rather than rotation, thus they are compressed and severely buckled under the oblique loading. Such inferior local deformation mode makes the 2nd order hierarchy easily to collapse, thus weaken the overall performance of 45º H-ReH. By contrast, when increases the cell-wall angle to 60º, the 2nd order struts exhibit an entirely different deformation mechanism. As shown in Figure 6.13 (b), the joints in inclined members exhibit significant twisting, and the connected 2nd order struts rotate in a compliant mode. As the angle between inclined and horizontal struts are higher, the 2nd order struts in 60º H-ReH have better resistance to compression but less resistance to the rotation. Therefore, the 2nd order struts follow the twisting of the joint and rotate in same direction, distinctive from the compression and buckling behaviors in 45º H-ReH. Considering the force response of both 45º and 60º H-ReHs (Figure 6.4 (a) and Figure 6.5 (a)), it is obvious that the joint twisting and struts compliant rotation mode retains the superiority of the 2nd order struts and provides better performance of H-ReH under the oblique loading condition. 124 Figure 6.12. DIC strain fields (major strain) of 60º R-ReH and H-ReH fabricated with soft material. (a) R-ReH and (b) H-ReH at displacement of 3.04 mm. (c) R-ReH and (d) H-ReH at displacement of 6.76 mm. Figure 6.13. Local deformation mechanism of 2nd order struts under the oblique loading. (a) Soft 45º H-ReH, and (b) soft 60º H-ReH. 125 6.3.3 Energy Absorption of ReHs under Oblique Loading To evaluate the oblique loading effect on energy absorption capacity of ReHs, the specific energy absorption capacity (Es) is calculated by Eqs (6.1): ∫ 𝐹 𝑑𝛿 𝐸𝑠 (𝜀) = (6.1) 𝑚 where F is the crushing force, δ is the crushing distance, and m is the mass of the ReHs. The maximum Es values of each ReHs at their densification or failure displacement, i.e. 𝐸𝑠𝑚𝑎𝑥 , are calculated and plotted in Figure 6.14. It is found that H-ReHs always have higher SEA over R- ReHs, confirming better structural performance of the 2nd order triangular hierarchy under the oblique loading condition. Specifically, as shown in Figure 6.14 (a), for soft 45º ReHs, it is observed that H-ReH has increased 𝐸𝑠𝑚𝑎𝑥 under the oblique loading over the axial one, while R- ReH exhibit a reduced 𝐸𝑠𝑚𝑎𝑥 under the oblique loading. The superiority of H-ReH under the oblique loading is due to the enhanced structural performance under the combined compressions and rotation effect endowed by the 45º 2nd order hierarchy, while R-ReH is vulnerable to the rotation effect thus weakened under the oblique loading due to its bending deformation mode. For rigid 45º ReHs, similar 𝐸𝑠𝑚𝑎𝑥 enhancement of H-ReH under oblique loading is observed, as shown in Figure 6.14 (b). In addition, the rigid R-ReH shows slightly higher 𝐸𝑠𝑚𝑎𝑥 values over axial one as well. This is due to the slightly higher fracture displacement of R-ReH under the oblique loading, as additional rotation effect slightly delays the global failure of the R-ReH from the complete fracture of rigid inclined members. When increases the cell-wall angle to 60º, the 𝐸𝑠𝑚𝑎𝑥 of H-ReH under the oblique loading becomes lower than the axial ones, as shown in Figure 6.14 (c). This is because the additional rotation effect under the oblique loading leads to the 126 twisting of joints and compliant rotation of the 2nd order struts. By contrast, the R-ReH is almost independent with the loading direction. As shown in Figure 6.14 (d), when switches from soft material to rigid materials, similar behavior are observed for H-ReH, while R-ReH exhibits a lower 𝐸𝑠𝑚𝑎𝑥 under the oblique loading due to the severe rotation effect. Figure 6.14. Specific energy absorption of H-ReHs and R-ReHs under oblique and axial loading: (a) soft 45º ReHs, (b) rigid 45º ReHs, (c) soft 60º ReHs, and (d) rigid 60º ReHs. 6.4 Conclusions In this study, the oblique and axial crushing behaviors of ReHs are characterized under the 127 quasi-static compressions. The material effect is investigated by using soft and rigid materials for the ReHs fabrication, respectively, while the structural effect is explored by tuning the cell-wall angles of ReHs as well. The mechanical responses, deformation mechanisms, and energy absorptions of ReHs are investigated. The major findings are summarized as follows: (1) Oblique loading has significant effects on the mechanical responses of ReHs. For soft 45º ReHs, the oblique loading condition reduces the force response of R-ReH, but increases the force response of H-ReH. In addition, the constitutive behavior of the fabrication material is found to strongly affect the oblique crushing behaviors of ReHs as well. For soft 45º ReHs, the R-ReH unexpectedly exhibit higher force response over H-ReH under both oblique and axial loading condition. When switch to rigid material, the H-ReHs is reversely superior to the R-ReH. What’s more, the cell-wall angle has obvious effect on the oblique crushing behaviors as well. When increases the cell-wall angle to 60º, the soft R- ReH becomes inferior to the H-ReH, on the contrary to the 45º cell-wall angle cases. (2) The oblique loading induces both NPR effect and rotation effect on the deformation of ReHs. Through DIC analyses, the oblique loading effect on the deformation mechanisms of ReHs are revealed. For R-ReHs, it is found that bending deformation modes on R-ReHs are material dependent. While soft R-ReH has severe stretching and distributed deformation on the inclined members, rigid R-ReH forms plastic hinges with much localized deformation and larger rotation. In addition, it is found that cell-wall angles have strong effects on the local deformation behavior of H-ReHs. While soft 45º H-ReH has 128 local struts undergo severe buckling behavior, soft 60º H-ReH has a distinctive joint twisting and compliant struts rotating behavior. (3) The specific energy absorption results of ReHs shows that H-ReHs will always have higher SEA over the corresponding R-ReH under the oblique loading. In addition, 45º H-ReHs will have higher SEA under oblique loading compared with axial loading, while for 60º H- ReHs the oblique loading will reduce the SEA. On the other hand, the SEA of R-ReHs under oblique loading is strongly dependent on the cell-wall angle design and the constitutive materials. Overall, these findings provide new understandings on the oblique crushing behaviors of ReHs, which benefits the design of next generation lightweight robust materials for crashworthiness applications. 129 CHAPTER 7. CONCLUSIONS AND FUTURE WORK 7.1 Conclusions The research work presented in this dissertation focuses on the design, additive manufacturing, mechanical properties, and deformation mechanisms of hierarchical re-entrant honeycombs (H- ReHs). The overarching objective of this research is to provide in-depth understanding of the hierarchical structure design and the underlying deformation mechanisms of H-ReHs in various loading conditions, thus establishing the design principles of next-generation lightweight materials and structures with improved and tailorable mechanical properties for various engineering applications. The key scientific conclusions are summarized as follows: (1) Developed a new type of lightweight cellular materials with unprecedented mechanical properties through the combination of structural hierarchy and negative Poisson’s ratio effect. A novel type of hierarchical re-entrant honeycombs (H-ReHs) have been successfully designed through combining the structural hierarchy and auxetic cellular configurations. Such complex geometries across multiple length scales are manufactured through powerful advanced 3D printing technique. We have illustrated, both experimentally and numerically, that the mechanical properties of H-ReHs are significantly enhanced (250% on SEA) in comparison to the R-ReHs, due to the stretching-dominated behavior and uniquely combined local deformation mechanisms. In addition, we have revealed the competition between the 1st order and the 2nd order hierarchy governed by the aspect ratio of the 2nd order triangular hierarchy, which strongly affect the local deformation mode and mechanical performance of H-ReHs. The mechanical properties of H-ReHs are 130 predictable by the scaling law and much improved at lower density region, which proofs the H-ReHs are a type of advanced lightweight structure. (2) Demonstrated the effect of cell-wall angle design on the mechanical performance of H-ReHs. In H-ReHs, the adaptive and reconfigurable 2nd order triangular hierarchy governs the mechanical properties of the material. We have validated that the elastic modulus and strength of H-ReHs are strongly dependent on the cell-wall angles, while R- ReHs are inert to that angle change. In addition, the NPR behavior and local deformation mode of H-ReHs can be significantly altered by the cell-wall angle change to achieve targeted performance. In short, we have demonstrated that cell-wall angle is one of the key design parameters to directly tune and control the performance of H-ReHs. (3) Revealed the structural effect on the dynamic crushing behavior of H-ReHs. The dynamic crushing behavior of H-ReHs is distinct from the quasi-static one, due to the additional structural strain rate effect. We have found the H-ReHs exhibit different strain rate sensitivity in the elastic and plastic regions, owing to the exclusive structural micro- inertia effect and the localized deformation modes in contrast to the R-ReHs. In addition, we have proved the effectiveness of 2nd order hierarchy of H-ReHs on the SEA at various strain rates (enhancement of 260% and 480% under quasi-static and dynamic loading conditions, respectively). (4) Validated the improved crushing behaviors of H-ReHs under the oblique loading condition. We have demonstrated the superior mechanical performance of H-ReHs over R-ReHs under the oblique loading (up to 430% SEA enhancement). We have found that 131 the cell-wall angle significantly affects the oblique loading crushing behavior of H-ReHs. With specific 45° cell-wall angle design, H-ReHs exceptionally exhibit better mechanical performance under oblique condition over the axial loading condition. In addition, it is discovered that the rigidity of constitutive material strongly alters the oblique compressive behavior, due to two distinctive local deformation mechanisms governing the oblique crushing behavior of H-ReHs. The research outcome of this study eliminates the design constraints of R-ReHs, elaborates the underlying deformation mechanisms governing the unprecedented mechanical performance of the H-ReHs, and provides a new paradigm for the future design of lightweight but robust cellular materials. 7.2 Future Work 7.2.1 3D-Printed Hybrid Hierarchical Re-entrant Honeycomb Up to recently, the rational design of lightweight hierarchical re-entrant honeycombs (H-ReHs) are mainly focused on devising and optimizing geometric arrangements to achieve unusual mechanical properties. While another route, which is through the rational design of spatially distributing materials with different properties within the H-ReHs, is much less explored [217]. The future goal is to fabricate multi-material H-ReHs with rational assignment of hard and soft materials through advanced 3D printing technique, thus uncover broadened mechanical performance of H-ReHs with tunable properties. The proposed multi-material H-ReHs design is shown in Figure 7.1 (a), with one outer section and one inner section for different material 132 assignment, respectively. The preliminary results are shown in Figure 7.1 (b). It is found that the multi-material design in H-ReHs can effectively tailor the mechanical performance. Figure 7.1. (a) Design of multi-material H-ReH divided into outer and inner sections for different material assignment. (b) Stress-strain responses of multi-material H-ReH with hard, original, and soft outer section. 7.2.2 Hierarchical Re-entrant Honeycomb Based Advanced Structures With the current understandings on the design and fundamental deformation mechanisms of H-ReHs in this dissertation, the H-ReHs have great potential as a new core material to be integrated into advanced structures, such as sandwich panels, thin-walled shell structures, and two-phase composite materials, as shown in Figure 7.2. 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