Energy Bounds for a Compressed Elastic Film on a Substrate

David P. Bourne, Sergio Conti*, Stefan Müller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, σ, and a measure of the bonding strength between the film and substrate, γ. We prove upper bounds on the minimum energy of the form σaγb and find that there are four different parameter regimes corresponding to different values of a and b and to different folding patterns of the film. In some cases, the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds.

Original languageEnglish
Pages (from-to)453-494
Number of pages42
JournalJournal of Nonlinear Science
Issue number2
Early online date17 Oct 2016
Publication statusPublished - Apr 2017


  • Branching
  • Energy scaling
  • Pattern formation
  • Thin-film elasticity

ASJC Scopus subject areas

  • Modelling and Simulation
  • General Engineering
  • Applied Mathematics


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