Analysis of a compressed thin film bonded to a compliant substrate: the energy scaling law. (English) Zbl 1421.74070
Summary: We consider the deformation of a thin elastic film bonded to a thick compliant substrate, when the (compressive) misfit is far beyond critical. We take a variational viewpoint – focusing on the total elastic energy, i.e., the membrane and bending energy of the film plus the elastic energy of the substrate – viewing the buckling of the film as a problem of energy-driven pattern formation. We identify the scaling law of the minimum energy with respect to the physical parameters of the problem, and we prove that a herringbone pattern achieves the optimal scaling. These results complement previous numerical studies, which have shown that an optimized herringbone pattern has lower energy than a number of other patterns. Our results are different, because (i) we make the scaling law achieved by the herringbone pattern explicit, and (ii) we give an elementary, ansatz-free proof that no pattern can achieve a better law.
MSC:
| 74K35 | Thin films |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74K20 | Plates |
References:
| [1] | Audoly, B., Boudaoud, A.: 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, 2401-2421 (2008a) · Zbl 1171.74349 · doi:10.1016/j.jmps.2008.03.003 |
| [2] | Audoly, B., Boudaoud, A.: Buckling of a stiff film bound to a compliant substrate—Part II: A global scenario for the formation of herringbone pattern. J. Mech. Phys. Solids 56, 2422-2443 (2008b) · Zbl 1171.74350 · doi:10.1016/j.jmps.2008.03.002 |
| [3] | Audoly, B., Boudaoud, A.: Buckling of a stiff film bound to a compliant substrate—Part III: Herringbone solutions at large buckling parameter. J. Mech. Phys. Solids 56, 2444-2458 (2008c) · Zbl 1171.74351 · doi:10.1016/j.jmps.2008.03.001 |
| [4] | Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells. Oxford University Press, Oxford (2010) · Zbl 1223.74001 |
| [5] | Bella, P., Kohn, R.V.: Wrinkles as the result of compressive stresses in an annular thin film. Commun. Pure Appl. Math., to appear · Zbl 1302.74105 |
| [6] | Ben Belgacem, H., Conti, S., DeSimone, A., Müller, S.: Rigorous bounds for the Föppl-von Karman theory of isotropically compressed plates. J. Nonlinear Sci. 10, 661-683 (2000) · Zbl 1015.74029 · doi:10.1007/s003320010007 |
| [7] | Ben Belgacem, H., Conti, S., DeSimone, A., Müller, S.: Energy scaling of compressed elastic films—three dimensional elasticity and reduced theories. Arch. Ration. Mech. Anal. 164, 1-37 (2002) · Zbl 1041.74048 · doi:10.1007/s002050200206 |
| [8] | 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) · doi:10.1038/30193 |
| [9] | Breid, D., Crosby, A.J.: Surface wrinkling behavior of finite circular plates. Soft Matter 5, 425-431 (2009) · doi:10.1039/b807820c |
| [10] | Cai, S., Breid, D., Crosby, A.J., Suo, Z., Hutchinson, J.W.: Periodic patterns and energy states of buckled films on compliant substrates. J. Mech. Phys. Solids 59, 1094-1114 (2011) · Zbl 1270.74126 · doi:10.1016/j.jmps.2011.02.001 |
| [11] | Chen, X., Hutchinson, J.W.: Herringbone buckling patterns of compressed thin films on compliant substrates. J. Appl. Mech. 71, 597-603 (2004a) · Zbl 1111.74362 · doi:10.1115/1.1756141 |
| [12] | Chen, X., Hutchinson, J.W.: A family of herringbone patterns in thin films. Scr. Mater. 50, 797-801 (2004b) · doi:10.1016/j.scriptamat.2003.11.035 |
| [13] | Choksi, R., Kohn, R.V., Otto, F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Commun. Math. Phys. 201, 61-79 (1999) · Zbl 1023.82011 · doi:10.1007/s002200050549 |
| [14] | Choksi, R., Conti, S., Kohn, R.V., Otto, F.: Ground state energy scaling laws during the onset and destruction of the intermediate state in a type-I superconductor. Commun. Pure Appl. Math. 61, 595-626 (2008) · Zbl 1142.82031 · doi:10.1002/cpa.20206 |
| [15] | Conti, S.: Branched microstructures: scaling and asymptotic self-similarity. Commun. Pure Appl. Math. 53, 1448-1474 (2000) · Zbl 1032.74044 · doi:10.1002/1097-0312(200011)53:11<1448::AID-CPA6>3.0.CO;2-C |
| [16] | Conti, S., Maggi, F.: Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187, 1-48 (2008) · Zbl 1127.74005 · doi:10.1007/s00205-007-0076-2 |
| [17] | Davidovitch, B., Schroll, R.D., Vella, D., Addia-Bedia, M., Cerda, E.: Prototypical model for tensional wrinkling in thin sheets. Proc. Natl. Acad. Sci. 108, 18227-18232 (2011) · Zbl 1355.74042 · doi:10.1073/pnas.1108553108 |
| [18] | Efimenko, K., Rackaitis, M., Manias, E., Vaziri, A., Mahadevan, L., Genzer, J.: Nested self-similar wrinkling patterns in skins. Nat. Mater. 4, 293-297 (2005) · doi:10.1038/nmat1342 |
| [19] | Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180, 183-236 (2006) · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7 |
| [20] | Genzer, J., Groenewold, J.: Soft matter with hard skin: from skin wrinkles to templating and material characterization. Soft Matter 2, 310-323 (2006) · doi:10.1039/b516741h |
| [21] | Gioia, G., Ortiz, M.: Delamination of compressed thin films. Adv. Appl. Mech. 33, 119-192 (1997) · Zbl 0930.74024 · doi:10.1016/S0065-2156(08)70386-7 |
| [22] | Huang, Z., Hong, W., Suo, Z.: Evolution of wrinkles in hard films on soft substrates. Phys. Rev. E 70, 030601(R) (2004) |
| [23] | Huang, Z., Hong, W., Suo, Z.: Nonlinear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53, 2101-2118 (2005) · Zbl 1176.74116 · doi:10.1016/j.jmps.2005.03.007 |
| [24] | Huang, R., Im, S.H.: Dynamics of wrinkle growth and coarsening in stressed thin films. Phys. Rev. E 74, 026214 (2006) · doi:10.1103/PhysRevE.74.026214 |
| [25] | Im, S.H., Huang, R.: Wrinkle patterns of anisotropic crystal films on viscoelastic substrates. J. Mech. Phys. Solids 56, 3315-3330 (2008) · Zbl 1171.74352 · doi:10.1016/j.jmps.2008.09.011 |
| [26] | Jagla, E.: Modeling the buckling and delamination of thin films. Phys. Rev. B 75, 085405 (2007) · doi:10.1103/PhysRevB.75.085405 |
| [27] | Jin, W., Sternberg, P.: Energy estimates for the von Karman model of thin-film blistering. J. Math. Phys. 42, 192-199 (2001) · Zbl 1028.74036 · doi:10.1063/1.1316058 |
| [28] | Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. 47, 405-435 (1994) · Zbl 0803.49007 · doi:10.1002/cpa.3160470402 |
| [29] | Kohn, R.V., Nguyen, H.: Tension-induced wrinkling in a confined film: the energy scaling law and the associated cascade, in preparation |
| [30] | Lin, P.-C., Yang, S.: Spontaneous formation of one-dimensional ripples in transit to highly ordered two-dimensional herringbone structures through sequential and unequal biaxial mechanical stretching. Appl. Phys. Lett. 90, 241903 (2007) · doi:10.1063/1.2743939 |
| [31] | Lobkovsky, A.E., Witten, T.A.: Properties of ridges in elastic membranes. Phys. Rev. E 55, 1577-1589 (1997) · doi:10.1103/PhysRevE.55.1577 |
| [32] | Mahadevan, L., Rica, S.: Self-organized origami. Science 307, 1740 (2005) · doi:10.1126/science.1105169 |
| [33] | Ni, Y., He, L., Liu, Q.: Modeling kinetics of diffusion-controlled surface wrinkles. Phys. Rev. E 84, 051604 (2011) · doi:10.1103/PhysRevE.84.051604 |
| [34] | Song, J., Jiang, H., Choi, W.M., Khang, D.Y., Huang, Y., Rogers, J.A.: An analytical study of two-dimensional buckling of thin films on compliant substrates. J. Appl. Phys. 103, 014303 (2008a) · doi:10.1063/1.2828050 |
| [35] | Song, J., Jiang, H., Liu, Z.J., Khang, D.Y., Huang, Y., Rogers, J.A., Lu, C., Koh, C.G.: Buckling of a stiff thin film on a compliant substrate in large deformation. Int. J. Solids Struct. 45, 3107-3121 (2008b) · Zbl 1169.74410 · doi:10.1016/j.ijsolstr.2008.01.023 |
| [36] | Song, J., Jiang, H., Huang, Y., Rogers, J.A.: Mechanics of stretchable inorganic electronic materials. J. Vac. Sci. Technol. A 27, 1107-1125 (2009) · doi:10.1116/1.3168555 |
| [37] | Sultan, E., Boudaoud, A.: The buckling of a swollen thin gel layer bound to a compliant substrate. J. Appl. Mech. 75, 051002 (2008) · doi:10.1115/1.2936922 |
| [38] | Vandeparre, H., Damman, P.: Wrinkling of stimuloresponsive surfaces: mechanical instability coupled to diffusion. Phys. Rev. Lett. 101, 124301 (2008) · doi:10.1103/PhysRevLett.101.124301 |
| [39] | Venkataramani, S.C.: Lower bounds for the energy in a crumpled elastic sheet—a minimal ridge. Nonlinearity 17, 301-312 (2004) · Zbl 1058.74038 · doi:10.1088/0951-7715/17/1/017 |
| [40] | Witten, T.A.: Stress focusing in elastic sheets. Rev. Mod. Phys. 79, 643-675 (2007) · Zbl 1205.74116 · doi:10.1103/RevModPhys.79.643 |
| [41] | Yin, J., Chen, X., Sheinman, I.: Anisotropic buckling patterns in spheroidal film/substrate systems and their implications in some natural and biological systems. J. Mech. Phys. Solids 57, 1470-1484 (2009) · Zbl 1371.74205 · doi:10.1016/j.jmps.2009.06.002 |
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