TY - JOUR

T1 - Bifurcation mechanism underlying echelon-mode formation

AU - Murota, Kazuo

AU - Ikeda, Kiyohiro

AU - Terada, Kenjiro

N1 - Funding Information:
The authors are grateful to Hisashi Okamoto and Makoto Matsumoto for discussion and mathematical information. The assistance in the numerical analysis by Mr. Asao Mizuki is appreciated. This work is partially supported by Japan Society for the Promotion of Science.

PY - 1999/3/12

Y1 - 1999/3/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0032661556&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032661556&partnerID=8YFLogxK

U2 - 10.1016/S0045-7825(98)00206-0

DO - 10.1016/S0045-7825(98)00206-0

M3 - Article

AN - SCOPUS:0032661556

SN - 0374-2830

VL - 170

SP - 423

EP - 448

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

IS - 3-4

ER -