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Research Papers

Bifurcation Diagrams for the Formation of Wrinkles or Creases in Soft Bilayers

[+] Author and Article Information
Lihua Jin

School of Engineering and Applied Sciences,
Kavli Institute for Nanobio Science
and Technology,
Harvard University,
Cambridge, MA 02138

Anesia Auguste

Department of Polymer Science and Engineering,
University of Massachusetts,
Amherst, MA 01003

Ryan C. Hayward

Department of Polymer Science and Engineering,
University of Massachusetts,
Amherst, MA 01003
e-mail: hayward@umass.edu

Zhigang Suo

School of Engineering and Applied Sciences,
Kavli Institute for Nanobio Science
and Technology,
Harvard University,
Cambridge, MA 02138
e-mail: suo@seas.harvard.edu

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 23, 2015; final manuscript received April 14, 2015; published online April 30, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(6), 061008 (Jun 01, 2015) (11 pages) Paper No: JAM-15-1158; doi: 10.1115/1.4030384 History: Received March 23, 2015; Revised April 14, 2015; Online April 30, 2015

Subject to compression, elastic materials may undergo bifurcation of various kinds. A homogeneous material forms creases, whereas a bilayer consisting of a stiff film and a compliant substrate forms wrinkles. Here, we show several new types of bifurcation behavior for bilayers consisting of films and substrates of comparable elastic moduli. Depending on the ratios of moduli and thicknesses of the two materials, the critical strain for the onset of creases can be either smaller or larger than that for the onset of wrinkles. When the critical strain for the onset of creases is lower than that of wrinkles, creases can be subcritical or supercritical. When the critical strain for the onset of wrinkles is lower than that of creases, wrinkles can further channel to creases at a strain much lower than the critical strain for the onset of creases in a homogeneous material. Experiments, conducted with bilayer polydimethylsiloxane (PDMS) structures subject to compressive loading, show that the different types of bifurcation behavior agree with the theoretical predictions.

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References

Genzer, J., and Groenewold, J., 2006, “Soft Matter With Hard Skin: From Skin Wrinkles to Templating and Material Characterization,” Soft Matter, 2(4), pp. 310–323. [CrossRef]
Chen, X., and Yin, J., 2010, “Buckling Patterns of Thin Films on Curved Compliant Substrates With Applications to Morphogenesis and Three-Dimensional Micro-Fabrication,” Soft Matter, 6(22), pp. 5667–5680. [CrossRef]
Li, B., Cao, Y. P., Feng, X. Q., and Gao, H. J., 2012, “Mechanics of Morphological Instabilities and Surface Wrinkling in Soft Materials: A Review,” Soft Matter, 8(21), pp. 5728–5745. [CrossRef]
Khang, D. Y., Jiang, H. Q., Huang, Y., and Rogers, J. A., 2006, “A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates,” Science, 311(5758), pp. 208–212. [CrossRef] [PubMed]
Stafford, C. M., Harrison, C., Beers, K. L., Karim, A., Amis, E. J., Vanlandingham, M. R., Kim, H. C., Volksen, W., Miller, R. D., and Simonyi, E. E., 2004, “A Buckling-Based Metrology for Measuring the Elastic Moduli of Polymeric Thin Films,” Nat. Mater., 3(8), pp. 545–550. [CrossRef] [PubMed]
Lin, P. C., Vajpayee, S., Jagota, A., Hui, C. Y., and Yang, S., 2008, “Mechanically Tunable Dry Adhesive From Wrinkled Elastomers,” Soft Matter, 4(9), pp. 1830–1835. [CrossRef]
Chan, E. P., Smith, E. J., Hayward, R. C., and Crosby, A. J., 2008, “Surface Wrinkles for Smart Adhesion,” Adv. Mater., 20(4), pp. 711–716. [CrossRef]
Lin, P. C., and Yang, S., 2009, “Mechanically Switchable Wetting on Wrinkled Elastomers With Dual-Scale Roughness,” Soft Matter, 5(5), pp. 1011–1018. [CrossRef]
Kim, J., Yoon, J., and Hayward, R. C., 2010, “Dynamic Display of Biomolecular Patterns Through an Elastic Creasing Instability of Stimuli-Responsive Hydrogels,” Nat. Mater., 9(2), pp. 159–164. [CrossRef] [PubMed]
Kucken, M., and Newell, A. C., 2004, “A Model for Fingerprint Formation,” Europhys. Lett., 68(1), pp. 141–146. [CrossRef]
Pauchard, L., and Couder, Y., 2004, “Invagination During the Collapse of an Inhomogeneous Spheroidal Shell,” Europhys. Lett., 66(5), pp. 667–673. [CrossRef]
Savin, T., Kurpios, N. A., Shyer, A. E., Florescu, P., Liang, H. Y., Mahadevan, L., and Tabin, C. J., 2011, “On the Growth and Form of the Gut,” Nature, 476(7358), pp. 57–62. [CrossRef] [PubMed]
Richman, D. P., Stewart, R. M., Hutchinson, J. W., and Caviness, V. S., 1975, “Mechanical Model of Brain Convolutional Development,” Science, 189(4196), pp. 18–21. [CrossRef] [PubMed]
Bayly, P. V., Taber, L. A., and Kroenke, C. D., 2014, “Mechanical Forces in Cerebral Cortical Folding: A Review of Measurements and Models,” J. Mech. Behav. Biomed. Mater., 29(S), pp. 568–581. [CrossRef] [PubMed]
Budday, S., Raybaud, C., and Kuhl, E., 2014, “A Mechanical Model Predicts Morphological Abnormalities in the Developing Human Brain,” Sci. Rep., 4, p. 5644. [CrossRef] [PubMed]
Lidmar, J., Mirny, L., and Nelson, D. R., 2003, “Virus Shapes and Buckling Transitions in Spherical Shells,” Phys. Rev. E, 68(5), p. 051910. [CrossRef]
Dumais, J., and Steele, C. R., 2000, “New Evidence for the Role of Mechanical Forces in the Shoot Apical Meristem,” J. Plant Growth Regul., 19(1), pp. 7–18. [CrossRef] [PubMed]
Liang, H. Y., and Mahadevan, L., 2009, “The Shape of a Long Leaf,” Proc. Natl. Acad. Sci. U.S.A., 106(52), pp. 22049–22054. [CrossRef] [PubMed]
Liang, H. Y., and Mahadevan, L., 2011, “Growth, Geometry, and Mechanics of a Blooming Lily,” Proc. Natl. Acad. Sci. U.S.A., 108(14), pp. 5516–5521. [CrossRef] [PubMed]
Armon, S., Efrati, E., Kupferman, R., and Sharon, E., 2011, “Geometry and Mechanics in the Opening of Chiral Seed Pods,” Science, 333(6050), pp. 1726–1730. [CrossRef] [PubMed]
Shipman, P. D., and Newell, A. C., 2004, “Phyllotactic Patterns on Plants,” Phys. Rev. Lett., 92(16), p. 168102. [CrossRef] [PubMed]
Dervaux, J., and Amar, M. B., 2008, “Morphogenesis of Growing Soft Tissues,” Phys. Rev. Lett., 101(6), p. 068101. [CrossRef] [PubMed]
Katifori, E., Alben, S., Cerda, E., Nelson, D. R., and Dumais, J., 2010, “Foldable Structures and the Natural Design of Pollen Grains,” Proc. Natl. Acad. Sci. U.S.A., 107(17), pp. 7635–7639. [CrossRef] [PubMed]
Yin, J., Cao, Z. X., Li, C. R., Sheinman, I., and Chen, X., 2008, “Stress-Driven Buckling Patterns in Spheroidal Core/Shell Structures,” Proc. Natl. Acad. Sci. U.S.A., 105(49), pp. 19132–19135. [CrossRef] [PubMed]
Hohlfeld, E., and Mahadevan, L., 2011, “Unfolding the Sulcus,” Phys. Rev. Lett., 106(10), p. 105702. [CrossRef] [PubMed]
Biot, M. A., 1963, “Surface Instability of Rubber in Compression,” Appl. Sci. Res., 12(2), pp. 168–182.
Chen, X., and Hutchinson, J. W., 2004, “Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates,” ASME J. Appl. Mech., 71(5), pp. 597–603. [CrossRef]
Audoly, B., and Boudaoud, A., 2008, “Buckling of a Stiff Film Bound to a Compliant Substrate—Part I: Formulation, Linear Stability of Cylindrical Patterns, Secondary Bifurcations,” J. Mech. Phys. Solids, 56(7), pp. 2401–2421. [CrossRef]
Song, J., Jiang, H., Liu, Z. J., Khang, D. Y., Huang, Y., Rogers, J. A., Lu, C., and Koh, C. G., 2008, “Buckling of a Stiff Thin Film on a Compliant Substrate in Large Deformation,” Int. J. Solids Struct., 45(10), pp. 3107–3121. [CrossRef]
Kim, D. H., Lu, N. S., Ma, R., Kim, Y. S., Kim, R. H., Wang, S. D., Wu, J., Won, S. M., Tao, H., Islam, A., Yu, K. J., Kim, T. I., Chowdhury, R., Ying, M., Xu, L. Z., Li, M., Chung, H. J., Keum, H., McCormick, M., Liu, P., Zhang, Y. W., Omenetto, F. G., Huang, Y. G., Coleman, T., and Rogers, J. A., 2011, “Epidermal Electronics,” Science, 333(6044), pp. 838–843. [CrossRef] [PubMed]
Keplinger, C., Sun, J. Y., Foo, C. C., Rothemund, P., Whitesides, G. M., and Suo, Z. G., 2013, “Stretchable, Transparent, Ionic Conductors,” Science, 341(6149), pp. 984–987. [CrossRef] [PubMed]
Kolle, M., Lethbridge, A., Kreysing, M., Baumberg, J. J., Aizenberg, J., and Vukusic, P., 2013, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,” Adv. Mater., 25(15), pp. 2239–2245. [CrossRef] [PubMed]
Bowden, N., Brittain, S., Evans, A. G., Hutchinson, J. W., and Whitesides, G. M., 1998, “Spontaneous Formation of Ordered Structures in Thin Films of Metals Supported on an Elastomeric Polymer,” Nature, 393(6681), pp. 146–149. [CrossRef]
Cerda, E., and Mahadevan, L., 2003, “Geometry and Physics of Wrinkling,” Phys. Rev. Lett., 90(7), p. 074302. [CrossRef] [PubMed]
Tanaka, T., Sun, S. T., Hirokawa, Y., Katayama, S., Kucera, J., Hirose, Y., and Amiya, T., 1987, “Mechanical Instability of Gels at the Phase-Transition,” Nature, 325(6107), pp. 796–798. [CrossRef]
Gent, A. N., and Cho, I. S., 1999, “Surface Instabilities in Compressed or Bent Rubber Blocks,” Rubber Chem. Technol., 72(2), pp. 253–262. [CrossRef]
Ghatak, A., and Das, A. L., 2007, “Kink Instability of a Highly Deformable Elastic Cylinder,” Phys. Rev. Lett., 99(7), p. 076101. [CrossRef] [PubMed]
Trujillo, V., Kim, J., and Hayward, R. C., 2008, “Creasing Instability of Surface-Attached Hydrogels,” Soft Matter, 4(3), pp. 564–569. [CrossRef]
Hong, W., Zhao, X. H., and Suo, Z. G., 2009, “Formation of Creases on the Surfaces of Elastomers and Gels,” Appl. Phys. Lett., 95(11), p. 111901. [CrossRef]
Wang, Q. M., Zhang, L., and Zhao, X. H., 2011, “Creasing to Cratering Instability in Polymers Under Ultrahigh Electric Fields,” Phys. Rev. Lett., 106(11), p. 118301. [CrossRef] [PubMed]
Cai, S. Q., Chen, D. Y., Suo, Z. G., and Hayward, R. C., 2012, “Creasing Instability of Elastomer Films,” Soft Matter, 8(5), pp. 1301–1304. [CrossRef]
Weiss, F., Cai, S., Hu, Y., Kang, M. K., Huang, R., and Suo, Z., 2013, “Creases and Wrinkles on the Surface of a Swollen Gel,” J. Appl. Phys., 114(7), p. 073507. [CrossRef]
Dervaux, J., Couder, Y., Guedeau-Boudeville, M. A., and Amar, M. B., 2011, “Shape Transition in Artificial Tumors: From Smooth Buckles to Singular Creases,” Phys. Rev. Lett., 107(1), p. 018103. [CrossRef] [PubMed]
Diab, M. Z., Zhang, T., Zhao, R. K., Gao, H. J., and Kim, K. S., 2013, “Ruga Mechanics of Creasing: From Instantaneous to Setback Creases,” Proc. R. Soc. London, Ser. A, 469(2157), p. 20120753. [CrossRef]
Hutchinson, J. W., 2013, “The Role of Nonlinear Substrate Elasticity in the Wrinkling of Thin Films,” Philos. Trans. R. Soc., A, 371(1993), p. 20120422. [CrossRef]
Wu, Z., Bouklas, N., and Huang, R., 2013, “Swell-Induced Surface Instability of Hydrogel Layers With Material Properties Varying in Thickness Direction,” Int. J. Solids Struct., 50(3–4), pp. 578–587. [CrossRef]
Wang, Q. M., and Zhao, X. H., 2014, “Phase Diagrams of Instabilities in Compressed Film-Substrate Systems,” ASME J. Appl. Mech., 81(5), p. 051004. [CrossRef]
Cao, Y. P., and Hutchinson, J. W., 2012, “Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers,” ASME J. Appl. Mech., 79(3), p. 031019. [CrossRef]
Jin, L., Chen, D., Hayward, R. C., and Suo, Z., 2014, “Creases on the Interface Between Two Soft Materials,” Soft Matter, 10(2), pp. 303–311. [CrossRef] [PubMed]
Chen, D. Y., Cai, S. Q., Suo, Z. G., and Hayward, R. C., 2012, “Surface Energy as a Barrier to Creasing of Elastomer Films: An Elastic Analogy to Classical Nucleation,” Phys. Rev. Lett., 109(3), p. 038001. [CrossRef] [PubMed]
Hohlfeld, E., and Mahadevan, L., 2012, “Scale and Nature of Sulcification Patterns,” Phys. Rev. Lett., 109(2), p. 025701. [CrossRef] [PubMed]
Chen, D., Jin, L., Suo, Z., and Hayward, R. C., 2014, “Controlled Formation and Disappearance of Creases,” Mater. Horiz., 1(2), pp. 207–213. [CrossRef]
Jin, L., Takei, A., and Hutchinson, J. W., “Mechanics of Wrinkle/Ridge Transitions in Thin Film/Substrate Systems,” J. Mech. Phys. Solids (submitted).
Riks, E., 1972, “The Application of Newton's Method to the Problem of Elastic Stability,” ASME J. Appl. Mech., 39(4), p. 1060. [CrossRef]
Pocivavsek, L., Dellsy, R., Kern, A., Johnson, S., Lin, B. H., Lee, K. Y. C., and Cerda, E., 2008, “Stress and Fold Localization in Thin Elastic Membranes,” Science, 320(5878), pp. 912–916. [CrossRef] [PubMed]
Cai, S. Q., Bertoldi, K., Wang, H. M., and Suo, Z. G., 2010, “Osmotic Collapse of a Void in an Elastomer: Breathing, Buckling and Creasing,” Soft Matter, 6(22), pp. 5770–5777. [CrossRef]
Ogden, R. W., 1997, Non-Linear Elastic Deformation, Dover Publications, New York.
Huang, Z. Y., Hong, W., and Suo, Z., 2005, “Nonlinear Analyses of Wrinkles in a Film Bonded to a Compliant Substrate,” J. Mech. Phys. Solids, 53(9), pp. 2101–2118. [CrossRef]
Jiang, H. Q., Khang, D. Y., Song, J. Z., Sun, Y. G., Huang, Y. G., and Rogers, J. A., 2007, “Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports,” Proc. Natl. Acad. Sci. U.S.A., 104(40), pp. 15607–15612. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 1

A bilayer subject to compression may form wrinkles or creases. (a) A bilayer in the undeformed state. The surface of the film is set as X2 = 0, and the axis of X2 goes toward the substrate. (b) The bilayer subject to homogeneous compression. (c) The formation of wrinkles with wavelength lwrinkle. (d) The formation of creases with spacing lcrease.

Grahic Jump Location
Fig. 2

The critical strain ɛc for the onset of wrinkles as a function of the wavelength for several ratios of shear moduli and thicknesses. For each curve, the critical strain reaches a minimum for wrinkles of certain wavelength.

Grahic Jump Location
Fig. 3

The critical conditions for the onset of wrinkles represented in the plane with the axes of Gf/Gs and Hs/Hf. (a) The contour plots of the minimal critical strain under the uniaxial stress conditions or the plane strain conditions. The curve marked by ɛmUni=0.44 and ɛmPE=0.35 represents the conditions when the critical strain for the onset of creases coincides with that of the onset of wrinkles. Above this curve, the critical strain for the onset of wrinkles is smaller than that of creases. Below this curve, the critical strain for the onset of wrinkles is larger than that of creases. (b) The corresponding nominal wavelengths are independent of the type of deformation.

Grahic Jump Location
Fig. 4

The formation of creases in bilayers of various ratios of moduli and thicknesses. (a) Below the solid curve, the critical strain for the onset of wrinkles is larger than that of creases. Creases in a bilayer can be subcritical or supercritical. (b) Bifurcation diagram of a supercritical crease, where the applied strain represents the loading parameter, and the depth of the crease represents the state of the system. (c) Bifurcation diagram of a subcritical crease. (d) and (e) Calculated bifurcation diagrams for bilayers of some ratios of moduli and thicknesses. Morphology of subcritical creases right after snapping under the snapping forward strain ɛF=0.36 for (f) Gf/Gs=1.4 and Hs/Hf=19 and (g) Gf/Gs=1.6 and Hs/Hf=2.

Grahic Jump Location
Fig. 5

Bilayers of various ratios of moduli and thicknesses exhibit various types of bifurcation. (a) and (b) The critical strain for the onset of creases is smaller than that for the onset of wrinkles. Point A: The creases are supercritical. Point B: The creases are subcritical and can coexist with the flat state at a strain smaller than the critical strain for the onset of creases. Such a strain for the coexistence of two states is known as Maxwell strain. (c)–(f) The critical strain for the onset of wrinkles is smaller than that for the onset of creases. Point C: The flat state and creases coexist at a strain ɛMaxwellfc even smaller than the critical strain for the onset of wrinkles. Point D: The wrinkles form at a certain strain and can coexist with the creases at a larger strain ɛMaxwellwc. The creases can also coexist with the flat state at strain ɛMaxwellfc. Point E: The wrinkles form at a certain strain and can coexist with the creases at a larger strain ɛMaxwellwc. Point F: The wrinkles form, and no creases are found in the range of the strain calculated. (g) The points for various bifurcation behaviors identified on the plane of the axes Gf/Gs and Hs/Hf.

Grahic Jump Location
Fig. 6

Experimental setup and the observations of three types of bifurcation behavior. (a) A prestretched elastomeric mounting layer is used to apply compression to a film–substrate bilayer. Both the film and substrate are made of PDMS. We fabricate film–substrate bilayers of various modulus ratios to observe different types of bifurcation. (b), (c), and (d) the optical micrographs display the top views, and the inset confocal micrographs show the side views (the scale bars in the insets are equal to 100 μm). The micrographs are taken at the applied compressive strains where the instability first appear. (b) Subcritical creases at the strain of 0.37. (c) Coexistence of creases and wrinkles at the strain of 0.29. (d) Wrinkles at the strain of 0.21.

Grahic Jump Location
Fig. 7

The minimal critical strain ɛm and its corresponding wavelength Lwrinklem/hf depend on the ratios of moduli and thicknesses of the film and substrate. The solid curves are obtained when both the film and substrate are modeled as neo-Hookean materials. The circles correspond to approximate solutions when both the film and substrate are modeled as linearly elastic materials, and the film is modeled as a von Karman plate. The approximate solutions are accurate only when the film is much stiffer and thinner than the substrate.

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