Stability and post-bifurcation of film-substrate systems

Andrew Akerson, Ryan S. Elliott

Research output: Contribution to journalArticlepeer-review

Abstract

Morphological instabilities in soft solids with free surfaces lead to an array of deformation modes including wrinkling, creasing, folding and ridge localization. While homogeneous systems tend to form creases, stiff films over soft substrates usually exhibit surface waves. Here, we look to analytically investigate this transition through the effects of film stiffness and finite thickness on the post-bifurcation stability of these surface waves. By considering both the film and substrate as compressible Neo-Hookean solids, we apply bifurcation theory and Lyaponov-Schmidt-Koiter asymptotics to produce a phase diagram of the surface wave stability over the parameter space. While earlier works have studied the effect of film-to-substrate stiffness ratios for thin films on deep substrates in the incompressible setting, we consider the additional effects of both finite film thickness and Poisson ratio. To investigate the further evolution of these surface waves, we turn to computational methods through finite-element simulations with bifurcation branch-following techniques. We see that as the unstable surface waves evolve, they eventually lead to the beginnings of crease formation. Thus, when the surface waves are unstable, we would expect snap-back or snap-through behaviour leading to creases.

Original languageEnglish (US)
Article number20220181
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume478
Issue number2264
DOIs
StatePublished - Aug 31 2022

Bibliographical note

Funding Information:
R.S.E. would like to acknowledge support from the École Polytechnique and its Laboratore de Mécanique des Solides (LMS). The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this work. www.msi.umn.edu .

Funding Information:
The work of R.S.E. was partially supported by the U.S. NSF grant no. CMMI-1462826. Acknowledgements

Publisher Copyright:
© 2022 The Author(s).

Keywords

  • Asymptotic analysis
  • Buckling
  • Elastic material
  • Finite Strain
  • Stability and bifurcation

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