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Foldable structures made of hydrogel bilayers

1 Computer Science Department, University of Verona, Strada le Grazie 15, 37134 Verona - Italy
2 SISSA - International School for Advanced Studies, via Bonomea 265, 34136 Trieste - Italy

We discuss self-folding of a thin sheet by using patterned hydrogel bilayers, which act as hinges connecting flat faces. Folding is actuated by heterogeneous swelling due to different cross-linking densities of the polymer network in the two layers. Our analysis is based on a dimensionally reduced plate model, obtained by applying a recently developed theory [1], which provides us with an explicit connection between (three-dimensional) material properties and the curvatures induced at the hinges. This connection offers a recipe for the fabrication and design of the bilayers, by providing the values of the cross-linking density of each layer that need to be imprinted during polymerization in order to produce a desired folded shape upon swelling.
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Keywords hydrogels; folding; bilayers; dimension reduction; Gamma-convergence; Kirchho plate theory

Citation: Virginia Agostiniani, Antonio DeSimone, Alessandro Lucantonio, Danka Lučić. Foldable structures made of hydrogel bilayers. Mathematics in Engineering, 2019, 1(1): 204-223. doi: 10.3934/Mine.2018.1.204


  • 1. Agostiniani V, Lucantonio A, Lučić D (2018) Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets. ESAIM Contr Optim Ca in press.
  • 2. Agostiniani V, DeSimone A (2017) Rigorous derivation of active plate models for thin sheets of nematic elastomers. Math Mech Solids in press.
  • 3. Agostiniani V, DeSimone A (2017) Dimension reduction via Γ-convergence for soft active materials. Meccanica 52: 3457–3470.    
  • 4. Klein Y, Efrati E, Sharon E (2007) Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315: 1116–1120.    
  • 5. Aharoni H, Sharon E, Kupferman R (2014) Geometry of thin nematic elastomer sheets. Phys Rev Lett 113: 257801.    
  • 6. Hamouche W, Maurini C, Vincenti A, et al. (2017) Multi-parameter actuation of a neutrally stable shell: a flexible gear-less motor. Proc R Soc A 473: 20170364.    
  • 7. Armon S, Efrati E, Sharon E, et al. (2011) Geometry and mechanics in the opening of chiral seed pods. Science 333: 1726–1730.    
  • 8. Dawson C, Vincent JFV, Rocca AM (1997) How pine cones open. Nature 290: 668.
  • 9. Hu Z, Zhang X, Li Y (1995) Synthesis and application of modulated polymer gels. Science 269: 525–527.    
  • 10. Ionov L (2011) Soft microorigami: self-folding polymer films. Soft Matter 7: 6786–6791.    
  • 11. Guo W, Li M, Zhou J (2013) Modeling programmable deformation of self-folding all-polymer structures with temperature-sensitive hydrogels. Smart Mater Struct 22: 115028.    
  • 12. Mao Y, Yu K, Isakov MS, et al. (2015) Sequential self-folding structures by 3D printed digital shape memory polymers. Sci Rep-UK 5: 13616.    
  • 13. Na JH, Evans AA, Bae J, et al. (2015) Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers. Adv Mater 27: 79–85.    
  • 14. Liu Y, Shaw B, Dickey MD, et al. (2017) Sequential self-folding of polymer sheets. Sci Adv 3: e1602417.    
  • 15. Hawkes W, An B, Benbernou NM, et al. (2010) Programmable matter by folding. P Natl Acad Sci USA 107: 12441–12445.    
  • 16. Yoon C, Xiao R, Park J, et al. (2014) Functional stimuli responsive hydrogel devices by self-folding. Smart Mater Struct 23: 094008.    
  • 17. Doi M (2009) Gel dynamics. J Phys Soc Jpn 78: 052001.    
  • 18. Schmidt B (2007) Plate theory for stressed heterogeneous multilayers of finite bending energy. J Math Pure Appl 88: 107–122.    
  • 19. Lucantonio A, Nardinocchi P, Stone HA (2014) Swelling dynamics of a thin elastomeric sheet under uniaxial pre-stretch. J Appl Phys 115: 083505.    
  • 20. Dai HH, Song Z (2011) Some analytical formulas for the equilibrium states of a swollen hydrogel shell. Soft Matter 7: 8473–8483.    
  • 21. Topsøe F (2007) Some bounds for the logarithmic function, In: Cho YJ, Kim JK, Dragomir SS Editors, Inequality theory and applications 4, New York: Nova Science Publishers, 137.
  • 22. DeSimone A, Teresi L (2009) Elastic energies for nematic elastomers. Eur Phys J E 29: 191–204.    
  • 23. DeSimone A (1999) Energetics of fine domain structures. Ferroelectrics 222: 275–284.    


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