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Analysis of a Compressed Thin Film Bonded to a Compliant Substrate: The Energy Scaling Law

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Abstract

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.

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Notes

  1. In giving the substrate energies of u 3 and w the same weight in (7) we have ignored a constant of order one; as discussed earlier, this is appropriate since we seek only the energy scaling law.

  2. The paper Audoly and Boudaoud (2008c) proposes the Miura-ori construction as a model for the herringbone pattern in the far-from-critical regime; in particular, it does not differentiate between the two. In our view, however, a key difference is that the Miura-ori construction involves sharp creases, while the herringbone pattern involves smooth wrinkles.

  3. Here and in the rest of the paper, a≳b means a≥Cb for some positive universal constant C. Similarly, a≲b means that a≤Cb for some constant C, and a∼b means a≲b and b≲a.

  4. The summary presented here focuses on the rescaled problem, with misfit η=1. For a brief summary using the original variables see Remark 2 at the end of this section.

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Acknowledgements

R.V.K. gratefully acknowledges partial support from NSF grants DMS-0807347 and OISE-0967140. H.-M.N. gratefully acknowledges partial support from NSF grant DMS-1201370.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Robert V. Kohn

  2. Department of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Hoai-Minh Nguyen

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  1. Robert V. Kohn
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  2. Hoai-Minh Nguyen
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Correspondence to Robert V. Kohn.

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Communicated by P. Newton.

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Kohn, R.V., Nguyen, HM. Analysis of a Compressed Thin Film Bonded to a Compliant Substrate: The Energy Scaling Law. J Nonlinear Sci 23, 343–362 (2013). https://doi.org/10.1007/s00332-012-9154-1

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  • DOI: https://doi.org/10.1007/s00332-012-9154-1

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