An Origami-inspired Structure With Graded Stiffness
I. INTRODUCTION
Section:
The origami pattern of the origami-inspired waterbomb thin-shell structure consists of a repeating pattern of base units, each with a six-crease vertex, that is, a vertex where six creases converge. When the two ends of a waterbomb thin-shell structure are folded along the axial direction, they can be expanded in the radial direction from the center. This origami structure can be used to improve the robot's environmental adaptability. Different from the conventional mechanical structure, waterbomb structures can change their shapes through reversible elastic deformation. Although they are lightweight, they have the characteristics of high specific stiffness, controllable deformation, and simple assembly. Therefore, thin-walled waterbomb structures have found novel applications in soft robot grippers, wearable robots, variable-diameter wheel drive robots, bionic robots, and floating aerial robots. 1–7 1. S. Li, J. J. Stampfli, H. J. Xu, E. Malkin, E. V. Diaz, D. Rus, and R. J. Wood, "A vacuum-driven origami "Magic-ball" soft gripper," in Proceedings of the 2019 International Conference on Robotics and Automation (ICRA) (IEEE, Montreal, QC, Canada, 2019), pp. 7401–7408. 2. M. A. Robertson, O. C. Kara, and J. Paik, "Soft pneumatic actuator-driven origami-inspired modular robotic 'pneumagami'," Int. J. Rob. Res 40, 72–85 (2020). 3. D.-Y. Lee, S.-R. Kim, J.-S. Kim, J.-J. Park, and K.-J. Cho, "Origami wheel transformer: A variable-diameter wheel drive robot using an origami structure," Soft Rob. 4, 163–180 (2017). https://doi.org/10.1089/soro.2016.0038 4. J.-Y. Lee, B. B. Kang, D.-Y. Lee, S.-M. Baek, W.-B. Kim, W.-Y. Choi, J.-R. Song, H.-J. Joo, D. Park, and K.-J. Cho, "Development of a multi-functional soft robot (SNUMAX) and performance in RoboSoft grand challenge," Front. Rob. AI 3, 63 (2016). https://doi.org/10.3389/frobt.2016.00063 5. H. Fang, Y. Zhang, and K. W. Wang, "Origami-based earthworm-like locomotion robots," Bioinspiration Biomimetics 12, 065003 (2017). https://doi.org/10.1088/1748-3190/aa8448 6. C. D. Onal, R. J. Wood, and D. Rus, "An origami-inspired approach to worm robots," IEEE/ASME Trans. Mechatron. 18, 430–438 (2013). https://doi.org/10.1109/tmech.2012.2210239 7. P. H. Le, Z. Wang, and S. Hirai, "Origami structure toward floating aerial robot," in Proceedings of the 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM) (IEEE, Busan, South Korea, 2015), pp. 1565–1569. Equipment functions can be effectively improved by combining cutting-edge origami skills with traditional mechanisms. 8 8. Y. Feng, K. Li, Y. Gao, H. Qiu, and J. Liu, "Design and optimization of origami-inspired orthopyramid-like core panel for load damping," Appl. Sci. 9, 4619 (2019). https://doi.org/10.3390/app9214619
In recent years, mechanical research related to waterbomb thin-shell structures has yielded numerous results. Feng et al. discovered the twist motion of tubular mechanical metamaterials based on waterbomb origami and analyzed the stiffness changes with the continuous twist motion. 9 9. H. Feng, J. Ma, Y. Chen, and Z. You, "Twist of tubular mechanical metamaterials based on waterbomb origami," Sci. Rep. 8, 9522 (2018). https://doi.org/10.1038/s41598-018-27877-1 Mukhopadhyay et al. developed a waterbomb tubular mechanical metamaterial with programmable deformation-related stiffness and shape modulation functions. 10 10. T. Mukhopadhyay, J. Ma, H. Feng, D. Hou, J. M. Gattas, Y. Chen, and Z. You, "Programmable stiffness and shape modulation in origami materials: Emergence of a distant actuation feature," Appl. Mater. Today 19, 100537 (2020). https://doi.org/10.1016/j.apmt.2019.100537 Gillman et al. studied the folding mode of the waterbomb structure and explored how the interaction between the stretching energy and the folding ability affects the design of a bistable structure. 11 11. A. Gillman, K. Fuchi, and P. R. Buskohl, "Truss-based nonlinear mechanical analysis for origami structures exhibiting bifurcation and limit point instabilities," Int. J. Solids Struct. 147, 80–93 (2018). https://doi.org/10.1016/j.ijsolstr.2018.05.011 Fonseca et al. studied the nonlinear dynamics of the waterbomb wheel structure and developed a one-degree-of-freedom reduced-order model system to describe the waterbomb wheel. 12 12. L. M. Fonseca, G. V. Rodrigues, M. A. Savi, and A. Paiva, "Nonlinear dynamics of an origami wheel with shape memory alloy actuators," Chaos, Solitons Fractals 122, 245–261 (2019). https://doi.org/10.1016/j.chaos.2019.03.033 Hanna et al. performed dynamic analysis on the waterbomb base and predicted bistable behavior through kinematic and potential energy analyses. 13 13. B. H. Hanna, J. M. Lund, R. J. Lang, S. P. Magleby, and L. L. Howell, "Waterbomb base: A symmetric single-vertex bistable origami mechanism," Smart Mater. Struct. 23, 094009 (2014). https://doi.org/10.1088/0964-1726/23/9/094009 Jiayao et al. combined kinematics and structural analysis to characterize the folding process of the waterbomb structure and theoretically derived the geometric parameter range of rigid origami motion and nonrigid structure deformation. 14 14. J. Ma, H. Feng, Y. Chen, D. Hou, and Z. You, "Folding of tubularwaterbomb," Research 2020, 1735081. https://doi.org/10.34133/2020/1735081 Bowen et al. established a dynamic model of the waterbomb base to simulate the self-folding behavior of the waterbomb base driven by magneto-active elastomers (MAEs). 15 15. L. Bowen, K. Springsteen, H. Feldstein, M. Frecker, T. W. Simpson, and P. von Lockette, "Development and validation of a dynamic model of magneto-active elastomer actuation of the origami waterbomb base," J. Mech. Rob. 7, 011010 (2015). https://doi.org/10.1115/1.4029290 Glugla et al. used two mechanically different single-layer photopolymers to construct a waterbomb base and demonstrated experimentally that the waterbomb base of this molding method has good load-bearing capacity. 16 16. D. J. Glugla, M. D. Alim, K. D. Byars, D. P. Nair, C. N. Bowman, K. K. Maute, and R. R. McLeod, "Rigid origami via optical programming and deferred self-folding of a two-stage photo-polymer," ACS Appl. Mater. Interfaces 8, 29658–29667 (2016). https://doi.org/10.1021/acsami.6b08981
As for the research on the stiffness and critical buckling load of thin-shell structures, Filipov et al. introduced a method of connecting rigid foldable origami tubes with a "zipper" method to effectively improve the stiffness of the system. 17 17. E. T. Filipov, T. Tachi, and G. H. Paulino, "Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials," Proc. Natl. Acad. Sci. U. S. A. 112, 12321–12326 (2015). https://doi.org/10.1073/pnas.1509465112 Berger et al. determined the material geometry needed to achieve the Hashin–Shtrikman upper bound of isotropic elastic stiffness. 18 18. J. B. Berger, H. N. G. Wadley, and R. M. McMeeking, "Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness," Nature 543, 533–537 (2017). https://doi.org/10.1038/nature21075 Bertoldi et al. determined the design principles for the reprogrammable stiffness, shape conversion, and other functions that appear in thin-wall mechanical metamaterials. 19 19. K. Bertoldi, V. Vitelli, J. Christensen, and M. van Hecke, "Flexible mechanical metamaterials," Nat. Rev. Mater. 2, 17066 (2017). https://doi.org/10.1038/natrevmats.2017.66 Ma et al. created and studied an energy-absorbing origami structure with periodic gradient stiffness based on the Miura-ori folding pattern. 20,21 20. J. Ma, J. Song, and Y. Chen, "An origami-inspired structure with graded stiffness," Int. J. Mech. Sci. 136, 134–142 (2018). https://doi.org/10.1016/j.ijmecsci.2017.12.026 21. H. Wang, D. Zhao, Y. Jin, M. Wang, Z. You, and G. Yu, "Study of collapsed deformation and energy absorption of polymeric origami-based tubes with viscoelasticity," Thin Wall Struct. 144, 106246 (2019). https://doi.org/10.1016/j.tws.2019.106246 Through energy analysis, Zhai et al. created a thin-walled structure that can selectively collapse along two paths with different stiffness. 22 22. Z. Zhai, Y. Wang, and H. Jiang, "Origami-inspired, on-demand deployable and collapsible mechanical metamaterials with tunable stiffness," Proc. Natl. Acad. Sci. U. S. A. 115, 2032–2037 (2018). https://doi.org/10.1073/pnas.1720171115 More mechanical metamaterials with special purposes can be designed by using the principle in this work. Martinez et al. increased the stiffness and anisotropy of elastomeric actuators by introducing a folding structure into the actuators. 23 23. R. V. Martinez, C. R. Fish, X. Chen, and G. M. Whitesides, "Elastomeric origami: Programmable paper-elastomer composites as pneumatic actuators," Adv. Funct. Mater. 22, 1376–1384 (2012). https://doi.org/10.1002/adfm.201102978 Filipov et al. improved the simplified bar and hinge model, allowing it to provide a realistic representation of the stiffness characteristics and deformed shapes of origami structures. 24 24. E. T. Filipov, K. Liu, T. Tachi, M. Schenk, and G. H. Paulino, "Bar and hinge models for scalable analysis of origami," Int. J. Solids Struct. 124, 26–45 (2017). https://doi.org/10.1016/j.ijsolstr.2017.05.028 Guo et al. studied the effect of texturing on the critical pressure of hyperelastic tubes under axial load and internal pressure. 25 25. Z. Guo, J. Gattas, S. Wang, L. Li, and F. Albermani, "Experimental and numerical investigation of bulging behaviour of hyper-elastic textured tubes," Int. J. Mech. Sci. 115-116, 665–675 (2016). https://doi.org/10.1016/j.ijmecsci.2016.07.026 Ma and You proposed a thin-walled energy-absorbing device that reduces the initial buckling force by introducing a pre-folded surface to form geometric imperfections. 26 26. J. Ma and Z. You, "Energy absorption of thin-walled square tubes with a prefolded origami pattern—Part I: Geometry and numerical simulation," Int. J. Appl. Mech. 81, 1003 (2013). https://doi.org/10.1115/1.4024405 Holmes found that buckling, wrinkling, folding, creasing, and snapping can be combined as a mechanism to give thin-walled mechanical metamaterials the ability to morph from one shape to another. 27 27. D. P. Holmes, "Elasticity and stability of shape changing structures," Curr. Opin. Colloid Interface Sci. 40, 118–137 (2019). https://doi.org/10.1016/j.cocis.2019.02.008 Lee et al. used the imperfection-sensitive characteristics of thin-walled tubes to guide the deformation process, thereby controlling the buckling behavior of thin-walled tubes. 28 28. T.-U. Lee, X. Yang, J. Ma, Y. Chen, and J. M. Gattas, "Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns," Int. J. Mech. Sci. 151, 322–330 (2019). https://doi.org/10.1016/j.ijmecsci.2018.11.005 Wang et al. analyzed the elastic buckling of thin-walled tubes and other structures based on Eringen's nonlocal elasticity theory and Timoshenko beam theory. 29 29. C. M. Wang, Y. Y. Zhang, S. S. Ramesh, and S. Kitipornchai, "Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory," J. Phys. D: Appl. Phys. 39, 3904–3909 (2006). https://doi.org/10.1088/0022-3727/39/17/029 Neves et al. proposed a method to introduce the possibility of critical load control into a topology optimization model. 30 30. M. M. Neves, H. Rodrigues, and J. M. Guedes, "Generalized topology design of structures with a buckling load criterion," Struct. Optim. 10, 71–78 (1995). https://doi.org/10.1007/bf01743533 Jafari Mehrabadi et al. studied the effects of different geometric parameters and different types of estimation effective material properties on the critical mechanical buckling of functionally graded nanocomposite plates. 31 31. S. Jafari Mehrabadi, B. Sobhani Aragh, V. Khoshkhahesh, and A. Taherpour, "Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes," Composites, Part B 43, 2031–2040 (2012). https://doi.org/10.1016/j.compositesb.2012.01.067 Wu and Chen studied the buckling behavior of single-walled carbon nanotubes subjected to combined hydrostatic pressure and axial compression. 32 32. C.-P. Wu and Y.-J. Chen, "A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression," Mech. Mater. 148, 103514 (2020). https://doi.org/10.1016/j.mechmat.2020.103514 Panedpojaman et al. proposed a method to calculate the elastic buckling load of the long axis of cellular columns with multiple end openings. 33 33. P. Panedpojaman, T. Thepchatri, and S. Limkatanyu, "Elastic buckling of cellular columns under axial compression," Thin Wall Struct. 145, 106434 (2019). https://doi.org/10.1016/j.tws.2019.106434 Wagner et al. analyzed the buckling characteristics of thin cylindrical shell structures manufactured through different design concepts such as electroplating, machining, and welding under axial compression. 34 34. H. N. R. Wagner, C. Hühne, and I. Elishakoff, "Probabilistic and deterministic lower-bound design benchmarks for cylindrical shells under axial compression," Thin Wall Struct. 146, 106451 (2020). https://doi.org/10.1016/j.tws.2019.106451 In general, these studies have made great contributions to the practical application of origami structures. However, there is still a lack of detailed mechanical analysis of waterbomb origami structures. The critical buckling load of axial compression (critical buckling load, in short) of waterbomb structures is the parameter required for the waterbomb structure to achieve radial deformation, and the radial stiffness when the waterbomb structure fully deploys is the parameter required for it to achieve load-bearing performance.
In this study, we investigate a series of flexible thin-shell structures with base units in the shapes of square, rectangle, and parallelogram. We also conduct finite element simulations and quasi-static axial radial compression testing to study the relationships of critical buckling load and radial stiffness with structural parameters. 35 35. L. Yuan, H. Shi, J. Ma, and Z. You, "Quasi-static impact of origami crash boxes with various profiles," Thin Wall Struct. 141, 435–446 (2019). https://doi.org/10.1016/j.tws.2019.04.028 In Sec. II, we present the construction of the waterbomb structures and the corresponding structural parameters. In Sec. III, we describe the numerical simulation method and corresponding experimental verification. In Sec. IV, we analyze and discuss the results in detail. In Sec. V, we present the conclusions.
II. GEOMETRIC MODELING
Section:
As shown in Fig. 1 , all the waterbomb structures studied in this paper have 27 base units arranged in a grid three units wide and nine units tall. Figures 1(a) and 1(b) are the crease diagrams of the waterbomb origami pattern and the end caps on both ends of the waterbomb structure. The solid lines and dotted lines indicate the mountain and valley creases, respectively. The waterbomb structure shown in Fig. 1(c) is formed by tessellation.
FIG. 1. The geometry of waterbomb thin-shell structures: (a) end cap of the square waterbomb structure, (b) crease diagram of the square waterbomb structure, (c) square waterbomb structure, (d) end cap of the rectangle waterbomb structure, (e) crease diagram of the rectangle waterbomb structure, (f) rectangle waterbomb structure, (g) end cap of the parallelogram waterbomb structure, (h) crease diagram of the parallelogram waterbomb structure, and (i) parallelogram waterbomb structure.
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Figures 1(a) –1(c) show waterbomb structures with square base units, where m and n are the number of horizontal and vertical units. A square base unit is 2l a wide, t a is the length of the triangle centerline spliced on both sides of the opened diagram, and t b is the distance from the center of the regular polygon in the end cap to each vertex. Figures 1(d) –1(f) show waterbomb structures with rectangle base units. A rectangle base unit is 2l a long and 2l b wide. Figures 1(g) –1(i) show waterbomb structures with parallelogram base units with the tilt angle α.
In addition, u and v are the structure thickness and the gap in the unit facet in the waterbomb structure design, respectively. As shown in Fig. 2 , v describes the amount of soft material used in the waterbomb structure viewed from a side. Table I summarizes the geometric parameters of waterbomb thin-shell structures formed with the three types of base units.
TABLE I. Geometric parameters of the three waterbomb structures.
| Type of base units | Geometric parameter |
|---|---|
| Square | m, n, l a , t a , t b , u, v |
| Rectangle | m, n, l a , l b , t a , t b , u, v |
| Parallelogram | m, n, l a , l b , t a , t b , α, u, v |
In total, 34 waterbomb origami structures with different shape parameters are designed in this study, including Z1–Z6 with square base units, J1–J11 with rectangle base units, and P1–P17 with parallelogram base units. For quantitative evaluation, the critical buckling load-to-weight ratio, Q, and the radial stiffness-to-weight ratio, G, are introduced to facilitate the study of the effects of the parameters. The geometric parameters are listed in Tables II–IV.
TABLE II. Structural parameters of the square waterbomb.
| Model | m | n | l a (mm) | l b (mm) | t a (mm) | t b (mm) | α (rad) | u (mm) | v (mm) | l a /l b | G (N/mm kg−1) | Q (N/kg) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Z1 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 1 | ⋯ | 53.6 | 2192.3 |
| Z2 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 1.5 | ⋯ | 51.3 | 2076.9 |
| Z3 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 2 | ⋯ | 49.0 | 2019.6 |
| Z4 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 2.5 | ⋯ | 46.5 | 1923.1 |
| Z5 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 3 | ⋯ | 42.1 | 1781.8 |
| Z6 | 3 | 9 | 12 | ⋯ | 5 | 11.5 | ⋯ | 1.5 | 3.5 | ⋯ | 38.5 | 1750.0 |
TABLE III. Structural parameters of the rectangle waterbomb.
| Model | m | n | l a (mm) | l b (mm) | t a (mm) | t b (mm) | α (rad) | u (mm) | v (mm) | l a /l b | G (N/mm kg−1) | Q (N/kg) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| J1 | 3 | 9 | 10.5 | 16 | 5 | 11.5 | ⋯ | 1.5 | 1 | 0.66 | 53.6 | 2192.3 |
| J2 | 3 | 9 | 11.5 | 15 | 5 | 11.5 | ⋯ | 1.5 | 1 | 0.77 | 58.2 | 2019.6 |
| J3 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 1 | 1.04 | 65.2 | 1727.3 |
| J4 | 3 | 9 | 14.5 | 12 | 5 | 11.5 | ⋯ | 1.5 | 1 | 1.21 | 54.8 | 1614.0 |
| J5 | 3 | 9 | 15 | 11.5 | 5 | 11.5 | ⋯ | 1.5 | 1 | 1.3 | 45.8 | 1607.8 |
| J6 | 3 | 9 | 16 | 10.5 | 5 | 11.5 | ⋯ | 1.5 | 1 | 1.52 | 42.7 | 1365.4 |
| J7 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 1.5 | 1.04 | 60.6 | 1320.8 |
| J8 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 2 | 1.04 | 51.2 | 1152.8 |
| J9 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 2.5 | 1.04 | 49.6 | 1054.5 |
| J10 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 3 | 1.04 | 44.2 | 962.3 |
| J11 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | ⋯ | 1.5 | 3.5 | 1.04 | 40.7 | 854.5 |
TABLE IV. Structural parameters of the parallelogram waterbomb.
| Model | m | n | l a (mm) | l b (mm) | t a (mm) | t b (mm) | α (rad) | u (mm) | v (mm) | l a /l b | G (N/mm kg−1) | Q (N/kg) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| P1 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.105 | 1.5 | 1 | 1 | 75.55 | 1519.23 |
| P2 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.209 | 1.5 | 1 | 1 | 78.98 | 1442.31 |
| P3 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 1 | 1 | 84.53 | 1365.38 |
| P4 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.419 | 1.5 | 1 | 1 | 74.86 | 1326.92 |
| P5 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.524 | 1.5 | 1 | 1 | 68.68 | 1269.23 |
| P6 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.628 | 1.5 | 1 | 1 | 65.25 | 1057.69 |
| P7 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 1.5 | 1 | 83.01 | 1132.08 |
| P8 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 2 | 1 | 82.14 | 900.00 |
| P9 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 2.5 | 1 | 76.92 | 826.92 |
| P10 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 3 | 1 | 71.39 | 769.23 |
| P11 | 3 | 9 | 13.5 | 13.5 | 5 | 11.5 | 0.314 | 1.5 | 3.5 | 1 | 65.14 | 730.77 |
| P12 | 3 | 9 | 10.5 | 16 | 5 | 11.5 | 0.314 | 1.5 | 1 | 0.66 | 61.8 | 1423.1 |
| P13 | 3 | 9 | 11.5 | 15 | 5 | 11.5 | 0.314 | 1.5 | 1 | 0.77 | 58.2 | 1529.4 |
| P14 | 3 | 9 | 13.5 | 13 | 5 | 11.5 | 0.314 | 1.5 | 1 | 1.04 | 62.3 | 1290.9 |
| P15 | 3 | 9 | 14.5 | 12 | 5 | 11.5 | 0.314 | 1.5 | 1 | 1.21 | 60.3 | 1333.3 |
| P16 | 3 | 9 | 15 | 11.5 | 5 | 11.5 | 0.314 | 1.5 | 1 | 1.3 | 50.4 | 1411.8 |
| P17 | 3 | 9 | 16 | 10.5 | 5 | 11.5 | 0.314 | 1.5 | 1 | 1.52 | 56.1 | 1442.3 |
III. SIMULATION AND EXPERIMENT
Section:
A. Simulation model
The explicit dynamics solver of the finite element software ABAQUS/Explicit is used to simulate the axial and radial compression processes of the waterbomb structure with different types of base units and to solve highly nonlinear problems. In order to accurately simulate the large deformation states of the waterbomb structures, S4R shell elements are used. To ensure mesh convergence, the mesh size is set to 2 mm. Considering the surface contact, the static friction coefficient and the dynamic friction coefficient are both set to 0.25. The types of soft and hard materials are shown in Fig. 3 , and the material properties are shown in Table V.
TABLE V. Material properties.
| Elastic | Poisson's | Tensile | Shear | Polymer | Percentage elongation | |
|---|---|---|---|---|---|---|
| Material | modulus (MPa) | ratio | strength (MPa) | modulus (MPa) | density (g/cm3) | at fracture (%) |
| VeroBule rubber | 2500 | 0.35 | 60 | 926 | 1.18–1.19 | 15–25 |
| Agilus resin | 1 × 1011 | 0.45 | 2.4 | 0.238 | 1.14–1.15 | 220–240 |
The simulation is carried out in two steps. In the first step, the critical axial buckling load is solved in the following manner. The waterbomb structure is placed between two rigid plates; one rigid plate remains fixed and the other gradually compresses the sample. The compression displacement is 45 mm. The simulation results are shown in Fig. 6 . In the second step, the radial stiffness is solved in the following manner. When fully compressed (that is, the waterbomb structure is compressed until the middle units are fully expanded), the rigid plate is set to be tangent to the expanded state of the waterbomb structure, and the type of contact is defined as point-to-surface contact. The sample is compressed upwards perpendicularly in the horizontal direction. The compression displacement is 34.5 mm. The simulation results are shown in Fig. 8 .
In terms of Q and G, which are used as evaluation indicators, 34 waterbomb samples are evaluated. Q and G are defined in the following equations:
where F b is the critical buckling load, M is the mass of the waterbomb structure, F is the maximum radial load, and s is the radial displacement.
B. Experimental verification
In order to verify the accuracy of the ABAQUS simulation method, the square waterbomb structure Z3, the rectangle waterbomb structure J8, and the parallelogram waterbomb structure P8 are used in the comparative experiment. Quasi-static axial load tests and radial load tests are carried out to verify the validity of the numerical model. The waterbombs are made of soft and hard materials by 3D printing. The soft and hard materials are Agilus soft rubber and VeroBule resin, respectively, which adhere well to each other. The models are in a scale of 1:1. This fabrication process is quick and accurate, and it is convenient because one-time molding does not require complex assembly. Figure 4(a) shows the fabrication process, Fig. 4(b) shows the process of removing the support material, and Fig. 5 shows a finished sample to be tested.
FIG. 4. Waterbomb thin-shell structure fabrication: (a) 3D printing and (b) support material removal.
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FIG. 5. Waterbomb thin-shell structure: (a) square waterbomb structure test sample Z3, (b) rectangle waterbomb structure test sample J8, and (c) parallelogram waterbomb structure test sample P8.
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The WDW-10D and WDW-2D electronic universal testers are used to perform quasi-static axial compression tests and radial compression tests on the waterbomb samples. The axial compression test process is shown in Fig. 6 . During the process, the waterbomb structure is placed between two 3 cm3 sample plates, and a compression distance of 45 mm is applied at a loading rate of 1 mm min−1. The entire test process is recorded.
FIG. 6. Axial compression process of the waterbomb structure: (a) square waterbomb structure test sample Z3, (b) rectangle waterbomb structure test sample J8, and (c) parallelogram waterbomb structure test sample P8.
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The radial compression test process is shown in Fig. 7 . During this process, the 3D printing mold is used to define the expansion state of the waterbomb structure, and the waterbomb structure is placed in the mold to make the waterbomb structure fully expand radially. Then, a compression distance of 34.5 mm is applied at a loading rate of 1 mm min−1. The entire test process is recorded.
FIG. 7. Experimental process of radial compression of the waterbomb structure: (a) the waterbomb structure is mounted, (b) the compression displacement is 0 mm, (c) the compression displacement reaches 17.25 mm, and (d) the compression displacement reaches 34.5 mm.
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Comparisons between the axial test results and the simulation results as well as between the radial test results and the simulation results are shown in Figs. 10 and 11 , respectively. The calculations show that the maximum errors between the simulation and experimental test results of the square, rectangle, and parallelogram waterbombs are 9.34%, 9.88%, and 9.76%, respectively. Therefore, the finite element method is considered suitable for this study. The main reason for these errors is the errors between the numerical model and sample dimensions due to inaccuracy in machining. Because our study focuses on the overall mechanical properties of the waterbomb structures, we do not explore these factors.
IV. RESULTS AND DISCUSSION
Section:
A. Comparison of three different basic units
We first compare the parallelogram waterbomb structure P8, the square waterbomb structure Z3, and the rectangle waterbomb structure J8. The axial compression process and the radial compression process of Z3, J8, and P8 are shown in Figs. 6 and 8 , respectively. The axial compression process and the radial compression process of P8 are shown in Figs. 6(c) and 7(c) , respectively. The parallelogram waterbomb structure changes with the compression displacement, which triggers the axial twist displacement mode. It can be seen from Fig. 9(a) that the twist displacement mode occurs in the support layers of the parallelogram waterbomb structure. As shown in Fig. 9(b) , the square waterbomb structure changes with the compression displacement, and the overall structure gradually expands radially outward with creases along the axial direction. Meanwhile, the rectangle waterbomb structure is not analyzed for presenting the same kinematic tendency as the square waterbomb structure.
FIG. 8. Radial compression process of the waterbomb structure: (a) square waterbomb structure test sample Z3, (b) rectangle waterbomb structure test sample J8, and (c) parallelogram waterbomb structure test sample P8.
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FIG. 9. Waterbomb structures: (a) the parallelogram waterbomb structure test sample P8 and (b) the square waterbomb structure test sample Z3.
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The force–displacement curves of the three models are shown in Figs. 10 and 11 . The numerical data are shown in Tables II–IV The radial stiffness-to-weight ratio of the square waterbomb structure Z3 is 49 N/mm·kg−1, and the critical buckling load-to-weight ratio is 2019.6 N/kg. The radial stiffness-to-weight ratio of the rectangular waterbomb structure J8 is 51.2 N/mm kg−1, and the critical buckling load-to-weight ratio is 1152.8 N/kg. The radial stiffness-to-weight ratio of the parallelogram waterbomb structure P8 is 82.14 N/mm kg−1, and the critical buckling load-to-weight ratio is 900 N/kg. It can be seen that the square waterbomb structure Z3 has the smallest radial stiffness-to-weight ratio and the largest critical buckling load-to-weight ratio, while the parallelogram waterbomb structure P8 has the largest radial stiffness-to-weight ratio and the smallest critical buckling load-to-weight ratio. The parallelogram waterbomb structure has the smallest critical buckling load-to-weight ratio. The radial stiffness-to-weight ratio and the critical buckling load-to-weight ratio of the parallelogram waterbomb structure P8 are 1.68 and 0.45 times those of Z3, respectively. This is because the axial stiffness of the overall structure is reduced when the tilt angle of the parallelogram waterbomb structure is adjusted. The radial stiffness of the structure is improved in its deployed state. Because the critical buckling load is a measure to evaluate the difficulty of axial deformation of the waterbomb structure and the radial stiffness is the best representation of the radial load-bearing capacity of the waterbomb structure in practical applications, we want the critical buckling load to be the smallest and the radial stiffness to be the greatest. Therefore, it can be seen that the performance of the parallelogram waterbomb structure P8 is the best.
FIG. 10. Experimental and numerical axial force–displacement curves of the three waterbomb structures.
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FIG. 11. Experimental and numerical radial force–displacement curves of the three waterbomb structures.
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B. Effect of side ratio
The axial compression process of J8 is shown in Fig. 6(b) . The radial compression process of J8 is shown in Fig. 8(b) . In order to analyze the effects of the side ratio, six types of rectangle waterbomb structure models (J1–J6) with different side ratios are established and analyzed. The specific parameters are shown in Table III. J1–J6 are simulated, and the numerical calculation results yield the trends of the radial stiffness-to-weight ratio and the critical buckling load-to-weight ratio with respect to the side ratio as shown in Figs. 12 and 13 , respectively.
As shown in Fig. 12 , we find that when the side ratio l a /l b < 1.04 and the other parameters are constant, the radial stiffness-to-weight ratio of the rectangle waterbomb structure increases with the increase of l a /l b . When the side ratio l a /l b increases from 0.66 to 1.04, the radial stiffness-to-weight ratio of the rectangle waterbomb structure increases from 53.59 to 65.22 N/mm·kg−1; when the side ratio l a /l b is equal to 1.04, the radial stiffness-to-weight ratio of the rectangle waterbomb structure reaches its peak value.
However, when l a /l b > 1.04, the radial stiffness-to-weight ratio of the rectangle waterbomb structure decreases as l a /l b increases. When the side ratio l a /l b increases from 0.66 to 1.04, the radial stiffness-to-weight ratio of the rectangle waterbomb structure decreases from 65.22 to 42.74 N/mm kg−1. However, as shown in Fig. 13 , regardless of whether l a /l b > 1.04 or l a /l b < 1.04, the critical buckling load-to-weight ratio decreases monotonously with increasing l a /l b . When the side ratio l a /l b increases from 0.66 to 1.52, the critical buckling load-to-weight ratio of the rectangle waterbomb structure decreases from 2192.3 to 1365.4 N/kg.
Comparing the parallelogram waterbomb structures P12–P17, we find that the ratio l a /l b is sensitive to the effects of the radial stiffness-to-weight ratio and the critical buckling load-to-weight ratio.
As shown in Figs. 12 and 13 , when the side ratio l a /l b is equal to 1.04, the radial stiffness-to-weight ratio of the parallelogram waterbomb structure P14 reaches the maximum value of 62.34 N/mm kg−1. The critical buckling load-to-weight ratio of P14 just reaches the minimum value of 1290.9 N/kg, representing P14 performing the best for its reasonable parameters matching.
C. Effect of tilt angle
Six types of parallelogram waterbomb structure models (P1–P6) with different tilt angles are established and analyzed to study the effect of tilt angle on the structure. The specific parameters are shown in Table IV. With all other parameters remaining constant, P1–P6 are simulated. The numerical calculation results show the changes in the ratio of critical buckling load-to-weight and the radial stiffness-to-weight ratio, as shown in Figs. 14 and 15 , respectively.
As shown in Fig. 14 , when all other initial variables remain constant, the radial stiffness-to-weight ratio first increases and then decreases with the increase of α. When the tilt angle α is equal to 0.314 rad, the radial stiffness-to-weight ratio is 84.53 N/mm kg−1. When the tilt angle α is equal to 0.628 rad, the radial stiffness-to-weight ratio reaches the minimum value of 65.25 N/mm kg−1.
Whereas, as shown in Fig. 15 , the critical buckling load-to-weight ratio decreases monotonically with the increase of the tilt angle α. When the tilt angle α increases from 0.105 to 0.628, the critical buckling load-to-weight ratio decreases from 1519.2 to 1057.7 N/kg. It can be seen that after the other initial parameters are determined, the appropriate tilt angle α can be determined such that the mechanical properties of the waterbomb structure become optimal.
D. Effect of gap in unit facet
In order to study the effect of the gap in the unit facet on the waterbomb structures, we compare the square waterbomb structures Z1–Z6, the rectangle waterbomb structures J3 and J7–J11, and the parallelogram waterbomb structures P3 and P7–P11. In total, 18 types of waterbomb structures are established and analyzed. The specific parameters of the models are shown in Tables II–IV. With other variables kept constant, the numerical calculation results yield the changes in the critical buckling load-to-weight ratio and the radial stiffness-to-weight ratio with the increase of the cell gap, as shown in Figs. 16 and 17 , respectively.
FIG. 17. Critical buckling load-to-weight ratio vs gap in the unit facet.
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The comparison shows that as the gap in the unit facet, v, increases, the critical buckling load-to-weight ratios and the radial stiffness-to-weight ratios of the square, rectangle, and parallelogram waterbomb structures all show decreasing trends.
As shown in Figs. 16 and 17 , when the gap in the unit facet v increases from 1 to 3.5 mm, the radial stiffness-to-weight ratio of the square waterbomb structure decreases from 53.59 to 38.46 N/mm kg−1; the radial stiffness-to-weight ratio of the rectangle waterbomb structure decreases from 65.22 to 40.71 N/mm kg−1; and the radial stiffness-to-weight ratio of the parallelogram waterbomb structure decreases from 84.53 to 65.14 N/mm kg−1. The critical buckling load-to-weight ratio of the square waterbomb structure decreases from 2192.3 to 1750 N/kg; the critical buckling load-to-weight ratio of the rectangle waterbomb structure decreases from 1727.3 to 854.5 N/kg; and the critical buckling load-to-weight ratio of the parallelogram waterbomb structure decreases from 1365.4 to 730.8 N/kg.
The main reason is that as v increases, the proportion of the flexible material used increases, which weakens the axial and radial rigidity of the waterbomb structures. These trends further imply the validity of the simulation results.
V. CONCLUSIONS
Section:
| (1) | Twist displacement mode along the axial direction is present in the parallelogram waterbomb structure under axial compression. By adjusting the parameters properly, the critical buckling load can be effectively reduced, while the radial stiffness-to-weight ratio is significantly improved: compared with the conventional square waterbomb structure, the critical buckling load decreases by 55.4% and the radial stiffness-to-weight ratio increases by 67.6%. In conclusion, the design method proposed in this paper can effectively improve the mechanical properties of waterbomb structures. | ||||
| (2) | When l a /l b < 1.04 and the other parameters remain constant, the radial stiffness-to-weight ratio of the rectangle waterbomb structure increases as l a /l b increases. When l a /l b > 1.04, the radial stiffness-to-weight ratio of the rectangle waterbomb structure decreases as l a /l b increases. Whether l a /l b is >1.04 or <1.04, the critical buckling load decreases with l a /l b . | ||||
| (3) | For the parallelogram waterbomb structure, when other initial parameters remain constant, the critical buckling load decreases monotonously as the tilt angle α increases, whereas the radial stiffness-to-weight ratio first increases and then decreases as the tilt angle α increases. It can be concluded that with other initial parameters having been determined, the appropriate α can be determined through analysis to optimize the performance of the waterbomb structure. | ||||
| (4) | As the gap in the unit facet, v, increases, the critical buckling loads and the radial stiffness-to-weight ratios of the square, rectangle, and parallelogram waterbomb structures all decrease. | ||||
The data that support the findings of this study are available within the article.
An Origami-inspired Structure With Graded Stiffness
Source: https://aip.scitation.org/doi/10.1063/5.0050396
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