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Parametric excitation of wrinkles in elastic sheets on elastic and viscoelastic substrates

  • Regular Article – Soft Matter
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Abstract

Thin elastic sheets supported on compliant media form wrinkles under lateral compression. Since the lateral pressure is coupled to the sheet’s deformation, varying it periodically in time creates a parametric excitation. We study the resulting parametric resonance of wrinkling modes in sheets supported on semi-infinite elastic or viscoelastic media, at pressures smaller than the critical pressure of static wrinkling. We find distinctive behaviors as a function of excitation amplitude and frequency, including (a) a different dependence of the dynamic wrinkle wavelength on sheet thickness compared to the static wavelength; and (b) a discontinuous decrease in the dominant wrinkle wavelength upon increasing excitation frequency at sufficiently large pressures. In the case of a viscoelastic substrate, resonant wrinkling requires crossing a threshold of excitation amplitude. The frequencies for observing these phenomena in relevant experimental systems are of the order of a kilohertz and above. We discuss experimental implications of the results.

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Notes

  1. The opposite limit, of sheet-dominated inertia, is analyzed in the Supplementary Material [27]. In this limit, we expect a crossover from \(\omega \sim \omega _\mathrm{bd} \sim q^2\) to \(\omega \sim \omega _\mathrm{ad} \sim q^{1/2}\)

  2. Note that at intermediate frequencies this kernel describes a more complex response, including imaginary (yet still time-reversible) terms.

  3. In the static limit (\(\omega =0\)), one recovers the result derived from the Boussinesq problem [31], \({\tilde{K}}(q,0)=2G|q|\).

References

  1. E. Cerda, L. Mahadevan, Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 074302 (2003)

    ADS  Google Scholar 

  2. J. Genzer, J. Groenewold, Soft matter with hard skin: From skin wrinkles to templating and material characterization. Soft Matter 2, 310–323 (2006)

    Article  ADS  Google Scholar 

  3. B. Davidovitch, R.D. Schroll, D. Vella, M. Adda-Bedia, E. Cerda, Prototypical model for tensional wrinkling in thin sheets. Proc. Natl. Acad. Sci. USA 108, 18227–18232 (2011)

    Article  ADS  Google Scholar 

  4. L. Pocivavsek, J. Pugar, R. O’Dea, S.-H. Ye, W. Wagner, E. Tzeng, S. Velankar, E. Cerda, Topography-driven surface renewal. Nat. Phys. 14, 948–953 (2018)

    Article  Google Scholar 

  5. L. Pocivavsek, S.-H. Yea, J. Pugar, E. Tzeng, E. Cerda, S. Velankar, W.R. Wagnera, Active wrinkles to drive self-cleaning: A strategy for anti-thrombotic surfaces for vascular grafts. Biomat. 192, 226–234 (2019)

    Article  Google Scholar 

  6. N.N. Nath, L. Pocivavsek, J.A. Pugar, Y. Gao, K. Salem, N. Pitre, R. McEnaney, S. Velankar, E. Tzeng, Dynamic luminal topography: A potential strategy to prevent vascular graft thrombosis. Front. Bioeng. Biotech. 8, 573400 (2020)

    Article  Google Scholar 

  7. G. Lin, W. Sun, P. Chen, Topography-driven delamination of thin patch adhered to wrinkling surface. Int. J. Mech. Sci. 178, 105622 (2020)

    Article  Google Scholar 

  8. X. Wen, S. Sun, P. Wu, Dynamic wrinkling of a hydrogel-elastomer hybrid microtube enables blood vessel-like hydraulic pressure sensing and flow regulation. Mater. Horiz. 7, 2150 (2020)

    Article  Google Scholar 

  9. D. Vella, J. Bico, A. Boudaoud, B. Roman, P.M. Reis, The macroscopic delamination of thin films from elastic substrates. Proc. Natl. Acad. Sci. USA 106, 10901–10906 (2009)

    Article  ADS  Google Scholar 

  10. H. Mei, C.M. Landis, R. Huang, Concomitant wrinkling and buckle-delamination of elastic thin films on compliant substrates. Mech. Mater. 43, 627–642 (2011)

    Article  Google Scholar 

  11. E. Hohfeld, B. Davidovitch, Sheet on a deformable sphere: Wrinkle patterns suppress curvature-induced delamination. Phys. Rev. E 91, 012407 (2015)

    Article  ADS  Google Scholar 

  12. O. Oshri, Y. Liu, J. Aizenberg, A.C. Balazs, Delamination of a thin sheet from a soft adhesive Winkler substrate. Phys. Rev. E 97, 062803 (2018)

    Article  ADS  Google Scholar 

  13. O. Oshri, Delamination of open cylindrical shells from soft and adhesive Winkler’s foundation. Phys. Rev. E 102, 033001 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  14. G.D. Bixler, B. Bhushan, Biofouling: Lessons from nature. Phil. Trans. R. Soc. A 370, 2381–2417 (2012)

    Article  ADS  Google Scholar 

  15. N. Sridhar, D.J. Srolovitz, Z. Suo, Kinetics of buckling of a compressed film on a viscous substrate. Appl. Phys. Lett. 78, 2482–2484 (2001)

    Article  ADS  Google Scholar 

  16. R. Huang, Z. Suo, Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys. 91, 1135–1142 (2002)

    Article  ADS  Google Scholar 

  17. R. Huang, Kinetic wrinkling of an elastic film on a viscoelastic substrate. J. Mech. Phys. Solids 53, 63–89 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  18. R. Vermorel, N. Vandenberghe, E. Villermaux, Impacts on thin elastic sheets. Proc. Roy. Soc. A 465, 823–842 (2009)

    Article  ADS  Google Scholar 

  19. N. Vandenberghe, L. Duchemin, Impact on floating membranes. Phys. Rev. E 93, 052801 (2016)

    Article  ADS  Google Scholar 

  20. F. Box, D. O’Kiely, O. Kodio, M. Inizan, A.A. Castrejón-Pita, D. Vella, Dynamics of wrinkling in ultrathin elastic sheets. Proc. Natl. Acad. Soc. USA 116, 20875–20880 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  21. M.A. Ghanem, X. Liang, B. Lydon, L. Potocsnak, T. Wehr, M. Ghanem, S. Hoang, S. Cai, N. Boechler, Wrinkles riding waves in soft layered materials. Adv. Mat. Interface 6, 1801609 (2019)

    Article  Google Scholar 

  22. O. Kodio, I.M. Griffiths, D. Vella, Lubricated wrinkles: Imposed constraints affect the dynamics of wrinkle coarsening. Phys. Rev. Fluid 2, 014202 (2017)

    Article  ADS  Google Scholar 

  23. J. Chopin, M. Dasgupta, A. Kudrolli, Dynamic wrinkling and strengthening of an elastic filament in a viscous fluid. Phys. Rev. Lett. 119, 088001 (2017)

    Article  ADS  Google Scholar 

  24. F. Box, O. Kodio, D. O’Kiely, V. Cantelli, A. Goriely, D. Vella, Dynamic buckling of an elastic ring in a soap film. Phys. Rev. Lett. 124, 198003 (2020)

  25. H. Vandeparre, S. Gabriele, F. Brau, C. Gay, K.K. Parker, P. Damman, Hierarchical wrinkling patterns. Soft Matter 6, 5751–5756 (2010)

    Article  ADS  Google Scholar 

  26. L.D. Landau, E.M. Lifshitz, Mechanics, 2nd edn. (Pergamon Press, Oxford, 1960), sect. V.27

  27. See Supplementary Material

  28. J. Groenewold, Wrinkling of plates coupled with soft elastic media. Physica A 298, 32–45 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  29. L.D. Landau, E.M. Lifshitz, Theory of Elasticity, 3rd edn. (Butterworth-Heinemann, Oxford, 1986), sect. III.24

  30. H. Lamb, On the propagation of tremors over the surface of an elastic body. Phil. Trans. A 203, 1–42 (1904)

    ADS  MATH  Google Scholar 

  31. L.D. Landau, E.M. Lifshitz, Theory of Elasticity, 3rd edn. (Butterworth-Heinemann, Oxford, 1986), sect. I.8

  32. F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud, P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators. Nat. Phys. 7, 56–60 (2011)

    Article  Google Scholar 

  33. F. Brau, P. Damman, H. Diamant, T.A. Witten, Wrinkle to fold transition: influence of the substrate response. Soft Matter 9, 8177–8186 (2013)

    Article  ADS  Google Scholar 

  34. D.A. Dillard, B. Mukherjee, P. Karnal, R.C. Batra, J. Frechette, A review of Winkler’s foundation and its profound influence on adhesion and soft matter applications. Soft Matter 14, 3669–3683 (2018)

    Article  ADS  Google Scholar 

  35. A. Sonn-Segev, A. Bernheim-Groswasser, H. Diamant, Y. Roichman, Viscoelastic response of a complex fluid at intermediate distances. Phys. Rev. Lett. 112, 088301 (2014)

    Article  ADS  Google Scholar 

  36. A.Y. Grosberg, J.-F. Joanny, W. Srinin, Y. Rabin, Scale-dependent viscosity in polymer fluids. J. Phys. Chem. B 120, 6383–6390 (2016)

    Article  Google Scholar 

  37. C. Bar-Haim, H. Diamant, Surface response of a polymer network: Semi-infinite network. Langmuir 36, 247–255 (2020)

    Article  Google Scholar 

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Acknowledgements

Helpful discussions with Benny Davidovitch, Oz Oshri, and Luka Pocivavsek are gratefully acknowledged.

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  1. School of Chemistry, Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv, 6997801, Israel

    Haim Diamant

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  1. Haim Diamant
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Diamant, H. Parametric excitation of wrinkles in elastic sheets on elastic and viscoelastic substrates. Eur. Phys. J. E 44, 78 (2021). https://doi.org/10.1140/epje/s10189-021-00085-y

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