Building blocks that govern spontaneous and programmed pattern formation in pre-compressed bilayers

Jiajia Shen*, Alberto Pirrera, Rainer Groh

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)
14 Downloads (Pure)


Surface wrinkling in stiff-film/soft-substrate bilayers is a common phenomenon in biological systems and is increasingly being exploited in thin-film technology. While the onset of surface wrinkling in end-compressed bilayers is well understood, questions remain with regards to the evolution of the wrinkling pattern in the intermediate and deep post-wrinkling regimes, especially when the substrate is strongly pre-compressed. Here, we explore the bifurcation landscape of end-compressed bilayers with strongly pre-compressed substrates, using hyperelastic, plane strain finite elements and generalised path-following algorithms. After bifurcating from a flat into a sinusoidally wrinkled state, bilayers undergo further n-tupling bifurcations into stable wrinkling patterns of longer wavelength whose periodicity n = {4, ..., 8} is a function of overall bilayer length. These five n-tupling wrinkling patterns are shown to be independent localisations of the deformation mode and are accordingly identified as stable `building blocks' that govern the intermediate post-wrinkling regime. Additional end-shortening into the deep post-wrinkling regime then leads to further period doubling and coalescence of the building blocks. Beyond a certain length threshold, a bilayer can form a combinatorial side-by-side arrangement of the five building blocks. In the limit of an infinitely long bilayer, this leads to the phenomenon known as spatial chaos with the emergence of an infinite set of possible wrinkling patterns. In reality, though, the precise side-by-side arrangement of the building blocks is governed by the initial conditions. We show that the morphological evolution of the wrinkling pattern can be programmed by a judicial placement of manufactured dents in the thin film, creating new manufacturing capabilities for textured bilayers.
Original languageEnglish
Article number20220173
Number of pages23
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2265
Early online date21 Sept 2022
Publication statusPublished - 28 Sept 2022

Bibliographical note

Funding Information:
This work was funded by the Leverhulme Trust through a Philip Leverhulme Prize awarded to R.M.J.G. R.M.J.G. is also funded by the Royal Academy of Engineering under the Research Fellowship scheme (RF/201718/17178). A.P. was funded by the Engineering and Physical Sciences Research Council (EPSRC) under their Research Fellowship scheme (EP/M013170/1). The support of all funders is gratefully acknowledged.

Publisher Copyright:
© 2022 The Authors.


  • Soft matter
  • bilayer wrinkling
  • bifurcations
  • pattern formation
  • spatial chaos
  • modal nudging


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