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

  • Zi ChenEmail author
  • Xiaomin Han
  • Huang Zheng


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.


Misfit strain Residual stress Poisson’s effect Helices Nanoribbon Actuator 

Mathematics Subject Classification (2010)

74B10 74G15 70C20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Salamon, N.J., Masters, C.B.: Bifurcation in isotropic thin film/substrate plates. Int. J. Solids Struct. 32, 473–481 (1995) CrossRefzbMATHGoogle Scholar
  2. 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) CrossRefzbMATHGoogle Scholar
  3. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hubbard, T., Wylde, J.: Residual strain and resultant deflection of surface micromachined structures. J. Vac. Sci. Technol. A 18, 734–737 (2000) CrossRefGoogle Scholar
  5. 5.
    Klein, C.A.: How accurate are Stoney’s equation and recent modifications. J. Appl. Phys. 88, 5487–5489 (2000) CrossRefGoogle Scholar
  6. 6.
    Hsueh, C.: Modeling of elastic deformation of multilayers due to residual stresses and external bending. J. Appl. Phys. 91, 9652–9656 (2002) CrossRefGoogle Scholar
  7. 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) CrossRefGoogle Scholar
  8. 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) CrossRefGoogle Scholar
  9. 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) CrossRefGoogle Scholar
  10. 10.
    Kim, S., Boyd, J.G., Mani, S.: Mechanical behavior of mismatch strain-driven microcantilever. Microelectron. J. 38, 371–380 (2007) CrossRefGoogle Scholar
  11. 11.
    Li, X.: Strain induced semiconductor nanotubes: from formation process to device applications. J. Phys. D, Appl. Phys. 41, 193001 (2008) CrossRefGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 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) CrossRefGoogle Scholar
  14. 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. 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) CrossRefzbMATHGoogle Scholar
  16. 16.
    Gong, X.: Controlling surface properties of polyelectrolyte multilayers by assembly pH. Phys. Chem. Chem. Phys. 15, 10459–10465 (2013) CrossRefGoogle Scholar
  17. 17.
    Stoney, G.G.: The tension of metallic films deposited by electrolysis. Proc. R. Soc. A 82, 172–175 (1909) CrossRefGoogle Scholar
  18. 18.
    Timoshenko, S.: Analysis of bi-metal thermostats. J. Opt. Soc. Am. 11, 233–255 (1925) CrossRefGoogle Scholar
  19. 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) CrossRefGoogle Scholar
  20. 20.
    Huang, S., Zhang, X.: Gradient residual stress induced elastic deformation of multilayer MEMS structures. Sens. Actuators A 134, 177–185 (2006) CrossRefGoogle Scholar
  21. 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) CrossRefGoogle Scholar
  22. 22.
    Chen, Z., Majidi, C., Srolovitz, D.J., Haataja, M.: Tunable helical ribbons. Appl. Phys. Lett. 98, 011906 (2011) CrossRefGoogle Scholar
  23. 23.
    Chouaieb, N., Goriely, A., Maddocks, J.H.: Helices. Proc. Natl. Acad. Sci. USA 103, 9398–9403 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, J., Feng, X., Wang, G., Yu, S.: Twisting of nanowires induced by anisotropic surface stresses. Appl. Phys. Lett. 92, 191901 (2008) CrossRefGoogle Scholar
  25. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Armon, S., Efrati, E., Kupferman, R., Sharon, E.: Geometry and mechanics in the opening of chiral seed pods. Science 333, 1726–1730 (2011) CrossRefGoogle Scholar
  27. 27.
    Gerbode, S.J., Puzey, J.R., McCormick, A.G., Mahadevan, L.: How to cucumber tendril coils and overwinds. Science 337, 1087–1091 (2012) CrossRefGoogle Scholar
  28. 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. 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. 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) CrossRefGoogle Scholar
  31. 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) CrossRefGoogle Scholar
  32. 32.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, Elmsford (1986) Google Scholar
  33. 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) CrossRefzbMATHGoogle Scholar
  34. 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) CrossRefGoogle Scholar
  35. 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) CrossRefGoogle Scholar
  36. 36.
    Knowles, J.K.: Linear Vector Spaces and Cartesian Tensors. Oxford University Press, New York (1998) zbMATHGoogle Scholar
  37. 37.
    Oda, R., Huc, I., Schmutz, M., Candau, S.J., MacKintosh, F.C.: Tuning bilayer twist using chiral counterions. Nature 399, 566–569 (1999) CrossRefGoogle Scholar
  38. 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. 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) CrossRefGoogle Scholar
  40. 40.
    Efrati, E., Irvine, W.T.M.: Orientation-dependent handedness and chiral design. Phys. Rev. X 4, 011003 (2014) Google Scholar
  41. 41.
    Chen, Z.: Geometric nonlinearity and mechanical anisotropy in strained helical nanoribbons. Nanoscale 6, 9443–9447 (2014) CrossRefGoogle Scholar
  42. 42.
    Chen, Z.: Shape transition and multi-stability of helical ribbons: a finite element method study. Arch. Appl. Mech. 85(3), 331–338 (2015) CrossRefGoogle Scholar
  43. 43.
    Seffen, K.A., Guest, S.D.: Pre-stressed morphing bistable and neutrally stable shells. J. Appl. Mech. 78, 011002 (2011) CrossRefGoogle Scholar
  44. 44.
    Lachenal, X., Weaver, P.M., Daynes, S.: Multi-stable composite twisting structure for morphing applications. Proc. R. Soc. Lond. 468, 1230–1251 (2012) CrossRefGoogle Scholar
  45. 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) CrossRefGoogle Scholar
  46. 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. 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) CrossRefGoogle Scholar
  48. 48.
    Giomi, L., Mahadevan, L.: Multi-stability of free spontaneously curved anisotropic strips. Proc. R. Soc. Lond. 468, 511–530 (2012) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Ge, Q., Qi, H.J., Dunn, M.L.: Active materials by four-dimension printing. Appl. Phys. Lett. 103, 131901 (2013) CrossRefGoogle Scholar
  50. 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) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Thayer School of EngineeringDartmouth CollegeHanoverUSA
  2. 2.Fujian Radio and TV UniversityFuzhouChina

Personalised recommendations