Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Hierarchical folding of elastic membranes under biaxial compressive stress

Nature Materials volume 10, pages 952–957 (2011)Cite this article

Subjects

Abstract

Mechanical instabilities that cause periodic wrinkling during compression of layered materials find applications in stretchable electronics1,2,3 and microfabrication4,5,6,7, but can also limit an application’s performance owing to delamination or cracking under loading8 and surface inhomogeneities during swelling9. In particular, because of curvature localization, finite deformations can cause wrinkles to evolve into folds. The wrinkle-to-fold transition has been documented in several systems, mostly under uniaxial stress10,11,12,13. However, the nucleation, the spatial structure and the dynamics of the invasion of folds in two-dimensional stress configurations remain elusive. Here, using a two-layer polymeric system under biaxial compressive stress, we show that a repetitive wrinkle-to-fold transition generates a hierarchical network of folds during reorganization of the stress field. The folds delineate individual domains, and each domain subdivides into smaller ones over multiple generations. By modifying the boundary conditions and geometry, we demonstrate control over the final network morphology. The ideas introduced here should find application in the many situations where stress impacts two-dimensional pattern formation.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Morphological transition through fold localization.
Figure 2: Formation of the initial network of folds and bifurcation of a fold.
Figure 3: Dynamic reorganization of stress fields and tip–tip interactions in a domain.
Figure 4: Hierarchical partitioning of space accompanied with terminal and segmental branching.
Figure 5: Geometry-dependent networking process.

Similar content being viewed by others

References

  1. Rogers, J. A., Someya, T. & Huang, Y. G. Materials and mechanics for stretchable electronics. Science 327, 1603–1607 (2010).

    Article  CAS  Google Scholar 

  2. Kim, R. H. et al. Waterproof AlInGaP optoelectronics on stretchable substrates with applications in biomedicine and robotics. Nature Mater. 9, 929–937 (2010).

    Article  CAS  Google Scholar 

  3. Hamers, R. J. Flexible electronic futures. Nature 412, 489–490 (2001).

    Article  CAS  Google Scholar 

  4. Koo, W. H. et al. Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles. Nature Photon. 4, 222–226 (2010).

    Article  CAS  Google Scholar 

  5. Efimenko, K. et al. Nested self-similar wrinkling patterns in skins. Nature Mater. 4, 293–297 (2005).

    Article  CAS  Google Scholar 

  6. Stafford, C. M. et al. A buckling-based metrology for measuring the elastic moduli of polymeric thin films. Nature Mater. 3, 545–550 (2004).

    Article  CAS  Google Scholar 

  7. Bowden, N., Brittain, S, Evans, A. G., Hutchinson, J. W. & Whitesides, G. M. Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer. Nature 393, 146–149 (1998).

    Article  CAS  Google Scholar 

  8. Kim, D. H. et al. Stretchable and foldable silicon integrated circuits. Science 320, 507–511 (2008).

    Article  CAS  Google Scholar 

  9. Tanaka, T. et al. Mechanical instability of gels at the phase-transition. Nature 325, 796–798 (1987).

    Article  CAS  Google Scholar 

  10. Leahy, B. D. et al. Geometric stability and elastic response of a supported nanoparticle film. Phys. Rev. Lett. 105, 058301 (2010).

    Article  Google Scholar 

  11. Brau, F. et al. Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators. Nature Phys. 7, 56–60 (2010).

    Article  Google Scholar 

  12. Reis, P. M., Corson, F., Boudaoud, A. & Roman, B. Localization through surface folding in solid foams under compression. Phys. Rev. Lett. 103, 045501 (2009).

    Article  CAS  Google Scholar 

  13. Pocivavsek, L. et al. Stress and fold localization in thin elastic membranes. Science 320, 912–916 (2008).

    Article  CAS  Google Scholar 

  14. Biot, M. A. Folding instability of a layered viscoelastic medium under compression. Proc. R. Soc. Lond. Ser. A 242, 444–454 (1957).

    Article  CAS  Google Scholar 

  15. Huang, R. & Im, S. H. Dynamics of wrinkle growth and coarsening in stressed thin films. Phys. Rev. E 74, 026214 (2006).

    Article  Google Scholar 

  16. Edmondson, S., Frieda, K., Comrie, J. E., Onck, P. R. & Huck, W. T. S. Buckling in quasi-2D polymers. Adv. Mater. 18, 724–728 (2006).

    Article  CAS  Google Scholar 

  17. Chen, X. & Hutchinson, J. W. Herringbone buckling patterns of compressed thin films on compliant substrates. J. Appl. Mech.-T Asme 71, 597–603 (2004).

    Article  Google Scholar 

  18. Yin, H. Z. et al. Buckling suppression of SiGe islands on compliant substrates. J. Appl. Phys. 94, 6875–6882 (2003).

    Article  CAS  Google Scholar 

  19. Price, N. J. & Cosgrove, J. W. Analysis of Geological Structures (Cambridge Univ. Press, 1990).

    Google Scholar 

  20. Zartman, J. J. & Shvartsman, S. Y. Unit operations of tissue development: Epithelial folding. Annu. Rev. Chem. Biomol. 1, 231–246 (2010).

    Article  CAS  Google Scholar 

  21. Holmes, D. P. & Crosby, A. J. Draping flms: A wrinkle to fold transition. Phys. Rev. Lett. 105, 038303 (2010).

    Article  Google Scholar 

  22. Kim, J., Yoon, J. & Hayward, R. C. Dynamic display of biomolecular patterns through an elastic creasing instability of stimuli-responsive hydrogels. Nature Mater. 9, 159–164 (2010).

    Article  Google Scholar 

  23. Hohlfeld, E. & Mahadevan, L. Unfolding the sulcus. Phys. Rev. Lett. 106, 105702 (2011).

    Article  Google Scholar 

  24. Huang, Z. Y., Hong, W. & Suo, Z. Nonlinear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53, 2101–2118 (2005).

    Article  CAS  Google Scholar 

  25. Shorlin, K. A., de Bruyn, J. R., Graham, M. & Morris, S. W. Development and geometry of isotropic and directional shrinkage-crack patterns. Phys. Rev. E 61, 6950–6957 (2000).

    Article  CAS  Google Scholar 

  26. Bohn, S., Pauchard, L. & Couder, Y. Hierarchical crack pattern as formed by successive domain divisions. I. Temporal and geometrical hierarchy. Phys. Rev. E 71, 046214 (2005).

    Article  CAS  Google Scholar 

  27. Skjeltorp, A. T. & Meakin, P. Fracture in microsphere monolayers studied by experiment and computer-simulation. Nature 335, 424–426 (1988).

    Article  Google Scholar 

  28. Couder, Y., Pauchard, L., Allain, C., Adda-Bedia, M. & Douady, S. The leaf venation as formed in a tensorial field. Eur. Phys. J. B 28, 135–138 (2002).

    Article  CAS  Google Scholar 

  29. Corson, F., Henry, H. & Adda-Bedia, M. A model for hierarchical patterns under mechanical stresses. Phil. Mag. 90, 357–373 (2010).

    Article  CAS  Google Scholar 

  30. Kranz, R. L. Crack–crack and crack–pore interactions in stressed granite. Int. J. Rock Mech. Min. 16, 37–47 (1979).

    Article  Google Scholar 

  31. Kim, P., Jeong, H. E., Khademhosseini, A. & Suh, K. Y. Fabrication of non-biofouling polyethylene glycol micro- and nanochannels by ultraviolet-assisted irreversible sealing. Lab. Chip 6, 1432–1437 (2006).

    Article  CAS  Google Scholar 

  32. Pizzi, A. & Mittal, K. L. Handbook of Adhesive Technology 2nd edn (M. Dekker, 2003).

    Google Scholar 

  33. van Dillen, T., Polman, A., van Kats, C. M. & van Blaaderen, A. Ion beam-induced anisotropic plastic deformation at 300 keV. Appl. Phys. Lett. 83, 4315–4317 (2003).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This work was supported by the Multidisciplinary University Research Initiative (MURI) program project funded through the Army Research Office, the National Research Foundation of Korea Grant (NRF-2009-352-D00034) and the Centre National de la Recherche Scientifique. M.A. is particularly grateful for the support offered by H.A.S. during his stay.

Author information

Author notes

  1. Pilnam Kim

    Present address: Present address: Department of Materials Science and Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea,

  2. Pilnam Kim and Manouk Abkarian: These authors contributed equally to this work

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

    Pilnam Kim & Howard A. Stone

  2. Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, Montpellier, F-34095, France

    Manouk Abkarian

  3. CNRS, Laboratoire Charles Coulomb UMR 5221, Montpellier, F-34095, France

    Manouk Abkarian

Authors
  1. Pilnam Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manouk Abkarian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Howard A. Stone
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

P.K., M.A. and H.A.S. designed the concepts. P.K. and M.A. carried out the experiments. P.K., M.A. and H.A.S. discussed and interpreted results. P.K., M.A. and H.A.S. wrote the paper.

Corresponding author

Correspondence to Howard A. Stone.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 1298 kb)

Supplementary Information

Supplementary Movie (AVI 8116 kb)

Supplementary Information

Supplementary Movie (AVI 11696 kb)

Supplementary Information

Supplementary Movie (AVI 397 kb)

Supplementary Information

Supplementary Movie (AVI 4155 kb)

Supplementary Information

Supplementary Movie (AVI 47633 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, P., Abkarian, M. & Stone, H. Hierarchical folding of elastic membranes under biaxial compressive stress. Nature Mater 10, 952–957 (2011). https://doi.org/10.1038/nmat3144

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nmat3144

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing