Structural Engineering and Mechanics

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Wrinkling of a homogeneous thin solid film deposited on a functionally graded substrate

  • Noroozi, Masoud (Faculty of Mechatronics, Islamic Azad University (Karaj Branch))
  • Received : 2019.06.13
  • Accepted : 2019.11.25
  • Published : 2020.04.25
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Abstract

Thin films easily wrinkle under compressive loading due to their small bending stiffness resulting from their tiny thickness. For a thin film deposited on a functionally graded substrate with non-uniform stiffness exponentially changes along the length span in this paper, the uniaxial wrinkling problem is solved analytically in terms of hyper-Bessel functions. For infinite, semi-infinite and finite length systems the wrinkling load and wrinkling wavenumber are determined and compared with those in literature. In comparison with a homogeneous substrate-bounded film in which the wrinkling pattern is uniform along the length span, for a functionally graded substrate-film system the wrinkles accumulate around the softer location of the functionally graded substrate. Therefore, the effective length of the film influenced by the wrinkles decreases, the amplitude of the wrinkles on softer regions of the functionally graded substrate grows and the wrinkling load of the functionally graded substrates with higher softening rate decreases more. The results of the current research are expected to be useful in science and technology of thin films and wrinkling of the structures especially living tissues.

Keywords

Acknowledgement

Supported by : Islamic Azad University

The research described in this paper was financially supported by the Islamic Azad University (Karaj Branch) under grant number 152228.

References

  1. Aberle, A.G. (2009), "Thin-film solar cells", Thin Solid Films, 517(17), 4706-4710. https://doi.org/10.1016/j.tsf.2009.03.056.
  2. Allen, H.G. (1969), Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, UK.
  3. Amar, M.B. and Bordner, A. (2017), "Mimicking cortex convolutions through the wrinkling of growing soft bilayers", J. Elasticity, 129(1-2), 213-238. https://doi.org/10.1007/s10659-017-9622-9.
  4. Birman, V. and Bert, C.W. (2004), "Wrinkling of composite-facing sandwich panels under biaxial loading", J. Sandw. Struct. Mater., 6(3), 217-237. https://doi.org/10.1177/1099636204033643.
  5. Cao, Y.P., Jia, F., Zhao, Y., Feng, X.Q. and Yu, S.W. (2012), "Buckling and post-buckling of a stiff film resting on an elastic graded substrate", Int. J. Solids Struct., 49(13), 1656-1664. https://doi.org/10.1016/j.ijsolstr.2012.03.004.
  6. Cerda, E., Ravi-Chandar, K. and Mahadevan, L. (2002), "Thin films: Wrinkling of an elastic sheet under tension", Nature, 419(6907), 579. https://doi.org/10.1038/419579b.
  7. Cerda, E. and Mahadevan, L. (2003), "Geometry and physics of wrinkling", Phys. Rev. Lett., 90(7), 074302. https://doi.org/10.1103/PhysRevLett.90.074302.
  8. Cerda, E. (2005), "Mechanics of scars", J. Biomech., 38(8), 1598-1603. https://doi.org/10.1016/j.jbiomech.2004.07.026.
  9. Chen, X. and Hutchinson, J.W. (2004), "Herringbone buckling patterns of compressed thin films on compliant substrates", J. Appl. Mech., 71(5), 597-603. https://doi.org/10.1115/1.1756141.
  10. Chen, L., Zhang, Y., Swaddiwudhipong, S. and Liu, Z. (2014), "Mimicking the pattern formation of fruits and leaves using gel materials", Struct. Eng. Mech., 50(5), 575-588. https://doi.org/10.12989/sem.2014.50.5.575.
  11. Chen, Z., Chen, W. and Song, J. (2017), "Buckling of a stiff thin film on an elastic graded compliant substrate", Proc. Royal Soc. A, 473(2208), 20170410. https://doi.org/10.1098/rspa.2017.0410.
  12. Clark, B.R., Pantoya, M.L., Hunt, E.M., Kelly, T.J., Allen, B.F., Heaps, R.J. and Daniels, M.A. (2015), "Synthesis and characterization of flexible, free-standing, energetic thin films", Surf. Coat. Tech., 284, 422-426. https://doi.org/10.1016/j.surfcoat.2015.05.048.
  13. Fu, C., Xu, F. and Huo, Y. (2018), "Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates", Int. J. Solids Struct., 132, 264-277. https://doi.org/10.1016/j.ijsolstr.2017.10.018.
  14. Genzer, J. and Groenewold, J. (2006), "Soft matter with hard skin: From skin wrinkles to templating and material characterization", Soft Matter, 2(4), 310-323. https://doi.org/10.1039/B516741H.
  15. Howarter, J.A. and Stafford, C.M. (2010), "Instabilities as a measurement tool for soft materials", Soft Matter, 6(22), 5661-5666. https://doi.org/10.1039/C0SM00365D.
  16. Hu, J., He, Y., Lei, J., Liu, Z. and Swaddiwudhipong, S. (2014), "Mechanical behavior of composite gel periodic structures with the pattern transformation", Struct. Eng. Mech., 50(5), 605-616. https://doi.org/10.12989/sem.2014.50.5.605.
  17. Jia, Z. and Li, T. (2016), "Failure mechanics of a wrinkling thin film anode on a substrate under cyclic charging and discharging", Extreme Mechanics Letters, 8, 273-282. https://doi.org/10.1016/j.eml.2016.03.006.
  18. Karki, S., Kim, H., Na, S.J., Shin, D., Jo, K. and Lee, J. (2016), "Thin films as an emerging platform for drug delivery", Asian Journal of Pharmaceutical Sciences, 11(5), 559-574. https://doi.org/10.1016/j.ajps.2016.05.004.
  19. Kim, T.Y., Puntel, E. and Fried, E. (2012), "Numerical study of the wrinkling of a stretched thin sheet", Int. J. Solids Struct., 49(5), 771-782. https://doi.org/10.1016/j.ijsolstr.2011.11.018.
  20. Kiryakova, V.S. (1993), Generalized Fractional Calculus and Applications, CRC press.
  21. Kudrolli, A. and Chopin, J. (2018), "Tension-dependent transverse buckles and wrinkles in twisted elastic sheets", Proc. Royal Soc. A, 474(2214), 20180062. https://doi.org/10.1098/rspa.2018.0062.
  22. Lecieux, Y. and Bouzidi, R. (2010), "Experimental analysis on membrane wrinkling under biaxial load-Comparison with bifurcation analysis", Int. J. Solids Struct., 47(18-19), 2459-2475. https://doi.org/10.1016/j.ijsolstr.2010.05.005.
  23. Lee, D., Triantafyllidis, N., Barber, J.R. and Thouless, M.D. (2008), "Surface instability of an elastic half space with material properties varying with depth", J. Mech. Phys. Solids, 56(3), 858-868. https://doi.org/10.1016/j.jmps.2007.06.010.
  24. Li, B., Cao, Y.P., Feng, X.Q. and Gao, H. (2011), "Surface wrinkling of mucosa induced by volumetric growth: theory, simulation and experiment", J. Mech. Phys. Solids, 59(4), 758-774. https://doi.org/10.1016/j.jmps.2011.01.010.
  25. Mahendran, M. and Jeevaharan, M. (1999), "Local buckling behaviour of steel plate elements supported by a plastic foam material", Struct. Eng. Mech., 7(5), 433-445. http://dx.doi.org/10.12989/sem.1999.7.5.433.
  26. Maugis, D. (2013), Contact, Adhesion and Rupture of Elastic Solids, (Vol. 130), Springer Science & Business Media, NY, USA.
  27. Niu, K. and Talreja, R. (1999), "Modeling of wrinkling in sandwich panels under compression", J. Eng. Mech., 125(8), 875-883. https://doi.org/10.1061/(ASCE)0733-9399(1999)125:8(875).
  28. Nolte, A.J., Chung, J.Y., Davis, C.S. and Stafford, C.M. (2017), "Wrinkling-to-delamination transition in thin polymer films on compliant substrates", Soft Matter, 13(43), 7930-7937. http://dx.doi.org/10.1039/C7SM01472D.
  29. Noroozi, M. and Jiang, L. (2012), "Buckling and wrinkling of a functionally graded material (FGM) thin film", Int. J. Appl. Mech., 4(02), 1250012. https://doi.org/10.1142/S1758825112500123.
  30. Pocivavsek, L., Dellsy, R., Kern, A., Johnson, S., Lin, B., Lee, K. Y.C. and Cerda, E. (2008), "Stress and fold localization in thin elastic membranes", Science, 320(5878), 912-916. https://doi.org/10.1126/science.1154069.
  31. Ratzersdorfer, J. (1936), Die Knickfestigkeit von Staben und Stabwerken, Springer-Verlag.
  32. Reddy, J.N. (2006), Theory and Analysis of Elastic Plates and Shells, (2nd Edition), CRC Press, FL, USA.
  33. Song, J. (2010), "Herringbone buckling patterns of anisotropic thin films on elastomeric substrates", Appl. Phys. Lett., 96(5), 051913. https://doi.org/10.1063/1.3309696.
  34. Veronese, G.P., Allegrezza, M., Canino, M., Centurioni, E., Ortolani, L., Rizzoli, R., Morandi, V. and Summonte, C. (2015), "Graphene as transparent conducting layer for high temperature thin film device applications", Sol. Energy Mater. Sol. Cells, 138, 35-40. https://doi.org/10.1016/j.solmat.2015.02.026.
  35. Vonach, W.K. and Rammerstorfer, F.G. (2001), "A general approach to the wrinkling instability of sandwich plates", Struct. Eng. Mech., 12(4), 363-376. https://doi.org/10.12989/sem.2001.12.4.363.
  36. Wang, Y.H., Tham, L.G. and Cheung, Y.K. (2005), "Beams and plates on elastic foundations: a review", Progress in Structural Engineering and Materials, 7(4), 174-182. https://doi.org/10.1002/pse.202.
  37. Wang, C.G., Liu, Y.P., Lan, L., Li, L. and Tan, H.F. (2016), "Post-wrinkling analysis of a torsionally sheared annular thin film by using a compound series method", Int. J. Mech. Sci., 110, 22-33. https://doi.org/10.1016/j.ijmecsci.2016.02.011.
  38. Xu, R., Li, D., Liu, W., Jiang, J., Liao, Y. and Wang, J. (2015a), "Modified nonlinear force density method for form-finding of membrane SAR antenna", Struct. Eng. Mech., 54(6), 1045-1059. http://dx.doi.org/10.12989/sem.2015.54.6.1045.
  39. Xu, F., Potier-Ferry, M., Belouettar, S. and Hu, H. (2015b), "Multiple bifurcations in wrinkling analysis of thin films on compliant substrates", Int. J. Nonlin. Mech., 76, 203-222. https://doi.org/10.1016/j.ijnonlinmec.2014.12.006.
  40. Yu, S., Sun, Y., Ni, Y., Zhang, X. and Zhou, H. (2016), "Controlled formation of surface patterns in metal films deposited on elasticity-gradient PDMS substrates", ACS Appl. Mater. Interfaces, 8(8), 5706-5714. https://doi.org/10.1021/acsami.5b12369.
  41. Zhang, Y., Zheng, J., Fang, C., Li, Z., Zhao, X., Li, Y., Ruan, X. and Dai, Y. (2018), "Enhancement of silicon-wafer solar cell efficiency with low-cost wrinkle antireflection coating of polydimethylsiloxane", Sol. Energy Mater. Sol. Cells, 181, 15-20. https://doi.org/10.1016/j.solmat.2017.10.004.
  42. Zhao, J., Guo, X. and Lu, L. (2017), "Controlled wrinkling analysis of thin films on gradient substrates", Appl. Math. Mech., 38(5), 617-624. https://doi.org/10.1007/s10483-017-2199-9.
  43. Zheng, Y., Li, G.Y., Cao, Y. and Feng, X.Q. (2017), "Wrinkling of a stiff film resting on a fiber-filled soft substrate and its potential application as tunable metamaterials", Extreme Mechanics Letters, 11, 121-127. https://doi.org/10.1016/j.eml.2016.12.002.