Bifurcation mechanism underlying echelon-mode formation

Kazuo Murota, Kiyohiro Ikeda, Kenjiro Terada

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


This paper presents a theory on the underlying mathematical mechanism of the echelon mode (a series of parallel short wrinkles that looks like a flight of stairs or wild geese arranged in formation) which has been observed ubiquitously with uniform materials, but which has long denied successful numerical simulations. It is shown by means of the group-theoretic bifurcation theory that the echelon mode formation can be explained as a recursive (secondary, tertiary, . . .) symmetry-breaking bifurcation if O(2) × O(2) is chosen as the underlying symmetry to model the local uniformity of materials. This implies, for example, that the use of periodic boundaries is essential to successfully realize the oblique stripe patterns and the subsequent echelon mode formation in numerical simulations. In fact, a recursive bifurcation analysis of a rectangular domain with periodic boundaries subject to uniform uniaxial compression yields various kinds of patterns, such as diamond, stripe and echelon modes, which are often observed for materials under shear.

Original languageEnglish
Pages (from-to)423-448
Number of pages26
JournalComputer Methods in Applied Mechanics and Engineering
Issue number3-4
Publication statusPublished - 1999 Mar 12


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