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Residual Stresses and Poisson’s Effect Drive Shape Formation and Transition of Helical Structures

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The Mechanics of Ribbons and Möbius Bands

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Abstract

Strained multilayer structures are extensively investigated because of their applications in microelectromechanical/nano-elecromechanical systems. Here we employ a finite element method (FEM) to study the bending and twisting of multilayer structures subjected to misfit strains or residual stresses. This method is first validated by comparing the simulation results with analytic predictions for the bending radius of a bilayer strip with given misfit strains. Then, the FEM simulations are used to study the deformation of a bilayer strip subjected to a certain residual stress to examine the influence of Poisson’s effect. As predicted by elasticity theory, a nearly purely twisted ribbon results for a given mis-orientation angle, although the residual stress only has one non-zero principal component. Our results further show that for the same Poisson’s ratio, a transition from a twisted ribbon to a nearly cylindrical helical shape can occur, either when the strip becomes wide and thin enough or when the driving force is large enough. The combined effects of the residual stress and the Poisson’s ratio are also examined. Our work demonstrates the effective use of finite element simulations in controllable design of strained multilayer structures, which have broad potential applications in NEMS, sensors, drug delivery, morphing structures, active materials, optoelectronics, and bio-inspired robotics.

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References

  1. Salamon, N.J., Masters, C.B.: Bifurcation in isotropic thin film/substrate plates. Int. J. Solids Struct. 32, 473–481 (1995)

    Article  MATH  Google Scholar 

  2. Suo, Z., Ma, E.Y., Gleskova, H., Wagner, S.: Mechanics of rollable and foldable film-on-foil electronics. Appl. Phys. Lett. 74, 1177–1179 (1999)

    Article  MATH  Google Scholar 

  3. Freund, L.B.: Substrate curvature due to thin film mismatch strain in the nonlinear deformation range. J. Mech. Phys. Solids 48, 1159–1174 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hubbard, T., Wylde, J.: Residual strain and resultant deflection of surface micromachined structures. J. Vac. Sci. Technol. A 18, 734–737 (2000)

    Article  Google Scholar 

  5. Klein, C.A.: How accurate are Stoney’s equation and recent modifications. J. Appl. Phys. 88, 5487–5489 (2000)

    Article  Google Scholar 

  6. Hsueh, C.: Modeling of elastic deformation of multilayers due to residual stresses and external bending. J. Appl. Phys. 91, 9652–9656 (2002)

    Article  Google Scholar 

  7. Feng, X., Huang, Y., Jiang, H., Ngo, D., Rosakis, A.J.: The effect of thin film/substrate radii on the Stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature. J. Mech. Mater. Struct. 1, 1041–1053 (2006)

    Article  Google Scholar 

  8. Zhang, L., Ruh, E., Grützmacher, D., Dong, L., Bell, D.J., Nelson, B.J., Schönenberger, C.: Anomalous coiling of SiGe/Si and SiGe/Si/Cr helical nanobelts. Nano Lett. 6, 1311–1317 (2006)

    Article  Google Scholar 

  9. Huang, S., Zhang, X.: Extension of the Stoney formula for film-substrate systems with gradient stress for MEMS applications. J. Micromech. Microeng. 16, 382–389 (2006)

    Article  Google Scholar 

  10. Kim, S., Boyd, J.G., Mani, S.: Mechanical behavior of mismatch strain-driven microcantilever. Microelectron. J. 38, 371–380 (2007)

    Article  Google Scholar 

  11. Li, X.: Strain induced semiconductor nanotubes: from formation process to device applications. J. Phys. D, Appl. Phys. 41, 193001 (2008)

    Article  Google Scholar 

  12. Singamaneni, S., LeMieux, M.C., Lang, H.P., Gerber, C., Lam, Y., Zauscher, S., Datskos, P.G., Lavrik, N.V., Jiang, H., Naik, R.R., Bunning, T.J., Tsukruk, V.V.: Bimaterial microcantilevers as a hybrid sensing platform. Adv. Mater. 20, 653–680 (2008)

    Article  Google Scholar 

  13. Li, W., Huang, G., Yan, H., Wang, J., Yu, Y., Hu, X., Wu, X., Mei, Y.: Fabrication and stimuli-responsive behavior of flexible micro-scrolls. Soft Matter 8, 7103–7107 (2012)

    Article  Google Scholar 

  14. Xu, D., Zhang, L., Dong, L., Nelson, B.J.: Nanorobotics for NEMS using helical nanostructures. In: Encyclopedia of Nanotechnology, pp. 1715–1721 (2012)

    Google Scholar 

  15. Childers, W.S., Anthony, N.R., Mehta, A.K., Berland, K.M., Lynn, D.G.: Phase networks of cross-\(\beta\) peptide assemblies. Langmuir 28, 6386–6395 (2012)

    Article  MATH  Google Scholar 

  16. Gong, X.: Controlling surface properties of polyelectrolyte multilayers by assembly pH. Phys. Chem. Chem. Phys. 15, 10459–10465 (2013)

    Article  Google Scholar 

  17. Stoney, G.G.: The tension of metallic films deposited by electrolysis. Proc. R. Soc. A 82, 172–175 (1909)

    Article  Google Scholar 

  18. Timoshenko, S.: Analysis of bi-metal thermostats. J. Opt. Soc. Am. 11, 233–255 (1925)

    Article  Google Scholar 

  19. Tsui, Y.C., Clyne, T.W.: An analytical model for predicting residual stresses in progressivelydeposited coatings part 1: planar geometry. Thin Solid Films 306, 23–33 (1997)

    Article  Google Scholar 

  20. Huang, S., Zhang, X.: Gradient residual stress induced elastic deformation of multilayer MEMS structures. Sens. Actuators A 134, 177–185 (2006)

    Article  Google Scholar 

  21. Pureza, J.M., Lacerda, M.M., De Oliveira, A.L., Fragalli, J.F., Zanon, R.A.S.: Enhancing accuracy to Stoney equation. Appl. Surf. Sci. 255, 6426–6428 (2009)

    Article  Google Scholar 

  22. Chen, Z., Majidi, C., Srolovitz, D.J., Haataja, M.: Tunable helical ribbons. Appl. Phys. Lett. 98, 011906 (2011)

    Article  Google Scholar 

  23. Chouaieb, N., Goriely, A., Maddocks, J.H.: Helices. Proc. Natl. Acad. Sci. USA 103, 9398–9403 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, J., Feng, X., Wang, G., Yu, S.: Twisting of nanowires induced by anisotropic surface stresses. Appl. Phys. Lett. 92, 191901 (2008)

    Article  Google Scholar 

  25. Majidi, C., Chen, Z., Srolovitz, D.J., Haataja, M.: Theory for the spontaneous bending of piezoelectric nanoribbons: mechanics, spontaneous polarization, and space charge coupling. J. Mech. Phys. Solids 58, 73–85 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Armon, S., Efrati, E., Kupferman, R., Sharon, E.: Geometry and mechanics in the opening of chiral seed pods. Science 333, 1726–1730 (2011)

    Article  Google Scholar 

  27. Gerbode, S.J., Puzey, J.R., McCormick, A.G., Mahadevan, L.: How to cucumber tendril coils and overwinds. Science 337, 1087–1091 (2012)

    Article  Google Scholar 

  28. Chen, Z., Majidi, C., Srolovitz, D.J., Haataja, M.: Continuum elasticity theory approach for spontaneous bending and helicity of ribbons induced by mechanical anisotropy. arXiv:1209.3321

  29. Armon, S., Aharoni, H., Moshe, M., Sharon, E.: Shape selection in chiral ribbons: from seed pods to supramolecular assemblies. Soft Matter (2014)

    Google Scholar 

  30. Sawa, Y., Ye, F., Urayama, K., Takigawa, T., Gimenez-Pinto, V., Selinger, R.L.B., Selinger, J.V.: Shape selection of twist-nematic-elastomer ribbons. Proc. Natl. Acad. Sci. USA 108, 6364–6368 (2011)

    Article  Google Scholar 

  31. Sawa, Y., Urayama, K., Takigawa, T., Gimenez-Pinto, V., Mbanga, B.L., Ye, F., Selinger, J.V., Selinger, R.L.B.: Shape and chirality transitions in off-axis twist nematic elastomer ribbons. Phys. Rev. E 88, 022502 (2013)

    Article  Google Scholar 

  32. Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, Elmsford (1986)

    Google Scholar 

  33. Guo, Q., Zheng, H., Chen, W., Chen, Z.: Finite element simulations on mechanical self-assembly of biomimetic helical structures. J. Mech. Med. Biol. 13, 1340018 (2013)

    Article  MATH  Google Scholar 

  34. Guo, Q., Mehta, A.K., Grover, M.A., Chen, W., Lynn, D.G., Chen, Z.: Shape selection and multi-stability in helical ribbons. Appl. Phys. Lett. 104, 211901 (2014)

    Article  Google Scholar 

  35. Guo, Q., Chen, Z., Li, W., Dai, P., Ren, K., Lin, J., Taber, L.A., Chen, W.: Mechanics of tunable helices and geometric frustration in biomimetic seashells. Europhys. Lett. 105, 64005 (2014)

    Article  Google Scholar 

  36. Knowles, J.K.: Linear Vector Spaces and Cartesian Tensors. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  37. Oda, R., Huc, I., Schmutz, M., Candau, S.J., MacKintosh, F.C.: Tuning bilayer twist using chiral counterions. Nature 399, 566–569 (1999)

    Article  Google Scholar 

  38. Guo, Q., Zheng, H., Chen, W., Chen, Z.: Modeling bistable behaviors in morphing structures through finite element simulations. Bio-Med. Mater. Eng. 24, 557–562 (2014)

    Google Scholar 

  39. Chen, Z., Guo, Q., Majidi, C., Chen, W., Srolovitz, D.J., Haataja, M.: Nonlinear geometric effects in bistable morphing structures. Phys. Rev. Lett. 109, 114302 (2012)

    Article  Google Scholar 

  40. Efrati, E., Irvine, W.T.M.: Orientation-dependent handedness and chiral design. Phys. Rev. X 4, 011003 (2014)

    Google Scholar 

  41. Chen, Z.: Geometric nonlinearity and mechanical anisotropy in strained helical nanoribbons. Nanoscale 6, 9443–9447 (2014)

    Article  Google Scholar 

  42. Chen, Z.: Shape transition and multi-stability of helical ribbons: a finite element method study. Arch. Appl. Mech. 85(3), 331–338 (2015)

    Article  Google Scholar 

  43. Seffen, K.A., Guest, S.D.: Pre-stressed morphing bistable and neutrally stable shells. J. Appl. Mech. 78, 011002 (2011)

    Article  Google Scholar 

  44. Lachenal, X., Weaver, P.M., Daynes, S.: Multi-stable composite twisting structure for morphing applications. Proc. R. Soc. Lond. 468, 1230–1251 (2012)

    Article  Google Scholar 

  45. Pirrera, A., Lachenal, X., Daynes, S., Weaver, P.M., Chenchiah, I.V.: Multi-stable cylindrical lattices. J. Mech. Phys. Solids 61, 2087–2107 (2013)

    Article  Google Scholar 

  46. Lachenal, X., Weaver, P.M., Daynes, S.: Influence of transverse curvature on the stability of pre-stressed helical structures. Int. J. Solids Struct. 468, 1230–1251 (2012)

    Google Scholar 

  47. Wissman, J., Finkenauer, L., Deseri, L., Majidi, C.: Saddle-like deformation in a dielectric elastomer actuator embedded with liquid-phase gallium-indium electrodes. J. Appl. Phys. 116, 144905 (2014)

    Article  Google Scholar 

  48. Giomi, L., Mahadevan, L.: Multi-stability of free spontaneously curved anisotropic strips. Proc. R. Soc. Lond. 468, 511–530 (2012)

    Article  MathSciNet  Google Scholar 

  49. Ge, Q., Qi, H.J., Dunn, M.L.: Active materials by four-dimension printing. Appl. Phys. Lett. 103, 131901 (2013)

    Article  Google Scholar 

  50. Hwang, G., Dockendorf, C., Bell, D., Dong, L., Hashimoto, H., Poulikakos, D., Nelson, B.: 3-D InGaAs/GaAs helical nanobelts for optoelectronic devices. Int. J. Optomechatron. 2, 88–103 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Thayer School of Engineering, Dartmouth College, Hanover, NH, 03755, USA

    Zi Chen & Xiaomin Han

  2. Fujian Radio and TV University, Fuzhou, China

    Huang Zheng

Authors
  1. Zi Chen
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  2. Xiaomin Han
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  3. Huang Zheng
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Correspondence to Zi Chen .

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Editors and Affiliations

  1. Aeropsace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota, USA

    Roger Fosdick

  2. Okinawa Institute of Science and Technol Mathematical Soft Matter Unit, Okinawa, Japan

    Eliot Fried

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Chen, Z., Han, X., Zheng, H. (2016). Residual Stresses and Poisson’s Effect Drive Shape Formation and Transition of Helical Structures. In: Fosdick, R., Fried, E. (eds) The Mechanics of Ribbons and Möbius Bands. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7300-3_16

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  • DOI: https://doi.org/10.1007/978-94-017-7300-3_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-7299-0

  • Online ISBN: 978-94-017-7300-3

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