Analysis of a compressed thin film bonded to a compliant substrate: The energy scaling law

Robert V. Kohn, Hoai Minh Nguyen

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)343-362
Number of pages20
JournalJournal of Nonlinear Science
Issue number3
StatePublished - Jun 2013


  • Compliant substrate
  • Energy scaling law
  • Foppl von Karman theory
  • Herringbone pattern
  • Thin film
  • Wrinkling

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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