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Echelon-Mode Formation

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Imperfect Bifurcation in Structures and Materials

Part of the book series: Applied Mathematical Sciences ((AMS,volume 149))

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Abstract

Bifurcation mechanism of the pattern formation in O(2) ×O(2)-invariant domains is studied. Diamond and stripe patterns are produced by direct bifurcations and echelon modes are engendered by secondary bifurcations. Bifurcating patterns are investigated based on experiments of soil, numerical simulations of sand, and image simulations of kaolin and steel. Fundamentals of group representation theory in Chap. 7 and group-theoretic bifurcation theory in Chap. 8 are the foundations of this chapter.

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Notes

  1. 1.

    The contents of this chapter are mostly based on Ikeda, Murota, and Nakano, 1994 [88]; and Ikeda et al., 2008 [98].

  2. 2.

    The echelon mode is found in various materials: soils (e.g., Ikeda, Murota, and Nakano, 1994 [88]), rocks (e.g., Davis, 1984 [33]), and metals (e.g., Bai and Dodd, 1992 [8] and Poirier, 1985 [158]). The cross-checker pattern, which is interpreted as an echelon symmetry in this chapter, is found in metals (e.g., Voskamp and Hoolox, 1998 [194]) and in the zebra patterns on the ocean floors (e.g., Nicolas, 1995 [142]). A self-similar pattern model was introduced by Archambault et al., 1993 [4].

  3. 3.

    In the Couette–Taylor flow , flows between two coaxial cylinders rotating with different angular velocities are investigated (cf., Taylor, 1923 [178]). For the Bénard convection, see, for example, Bénard, 1900 [11]; Chandrasekhar, 1961 [21]; and Koschmieder, 1966 [116], 1993 [118]. See also, for example, Drazin and Reid, 1981 [40] and Okamoto and Shoji, 2001 [149] for hydrodynamic stability.

  4. 4.

    See, for example, Schaeffer, 1980 [170]; Iooss, 1986 [100]; Bakker, 1991 [9]; Crawford and Knobloch, 1991 [31]; Chossat and Iooss, 1994 [25]; Chossat, 1994 [24]; Iooss and Adelmeyer, 1998 [101]; Moehlis and Knobloch, 2000 [134]; and Rabinovich, Ezersky, and Weidman, 2000 [161].

  5. 5.

    The group SO(2) ×O(2) acts as rotations about the axis of the cylindrical domain, translations in the axial direction and upside-down reflection, whereas group \({\mathrm {SO}}(2)\times {\mathbb {Z}}_{2}\) lacks the translational symmetry.

  6. 6.

    This is the case, for example, with the triaxial compression test on a cylindrical soil specimen .

  7. 7.

    We use the notation r(ψ) for any \(\psi \in \mathbb {R}\). For \(\psi \in \mathbb {R}\) not in the range of 0 ≤ ψ < 1, we define r(ψ) via periodic extension; e.g., r(2.3) = r(0.3) and r(−1.3) = r(−2.0 + 0.7) = r(0.7). Similarly for \(\tilde r(\tilde \psi )\).

  8. 8.

    The bifurcation analysis of an \({\mathrm {OB}}^{\pm }_{n{\tilde n}}\)-equivariant system is conducted in Sect. 16.8.2 and that of a D∞∞-equivariant system in Sect. 16.8.3.

  9. 9.

    D1 = 〈σ y σ z〉 denotes the half-rotation symmetry about the center of the specimen: the x-axis.

  10. 10.

    The formation of a shear band or a series of parallel shear bands is ascribed to a direct bifurcation in plastic bifurcation theory. This is called the shear-band mode bifurcation . See, for example, Hill and Hutchinson, 1975 [67] for this theory and Vardoulakis, Goldscheider, and Gudehus, 1978 [190] for its application to soil.

  11. 11.

    A uniaxial compression test on kaolin clay was conducted to obtain an image of a deformation pattern. The kaolin clay is suited for the visual observation of deformation patterns because of the geometrical characteristics of its grains that display optical anisotropy (Morgenstern and Tchalenko, 1967 [135]).

  12. 12.

    To make the representation (16.30) unique, we put A n0 = B n0 and C n0 = D n0 for n = 0,  1, …; and \(A_{0 \tilde {n}}=B_{0 \tilde {n}}\) and \(C_{0 \tilde {n}}=-D_{0 \tilde {n}}\) for \(\tilde n=0,1,\ldots \); in particular, C 00 = D 00 = 0.

  13. 13.

    This image is the 128×128 pixel digitized data obtained by an image scanner from the domain shown at the left of Fig. 16.14.

  14. 14.

    An endurance test on a steel ball bearing was conducted to obtain an image of a deformation pattern.

  15. 15.

    A fine angular, siliceous sand (Hostun RF) specimen of 164.0 mm × 173.0 mm × 35.4 mm was tested by the plane strain compression apparatus; the false relief stereophotogrammetry method was used to digitize the displacement fields of the side of the specimen deforming under load (Desrues and Viggiani, 2004 [36]).

  16. 16.

    These strain fields were obtained by processing the digitized data offered by J. Desrues for the study in Ikeda et al., 2008 [98].

  17. 17.

    The Fourier series (16.43) employed here is different from the series (16.30) that is used to detect stripe patterns.

  18. 18.

    Mode jumping means a sudden and dynamic shift to a different wave number.

  19. 19.

    Details of the numerical procedure are given in Ikeda, Yamakawa, and Tsutsumi, 2003 [99].

  20. 20.

    Closely located bifurcation points appear extensively in the bifurcation of materials, and are called a clustered bifurcation point or a point of accumulation (Hill and Hutchinson, 1975 [67]).

  21. 21.

    The column pattern, for example, is observed for the polygonal columns of layered basalt in Giant’s Causeway.

  22. 22.

    See Yamakawa, Hashiguchi, and Ikeda, 2010 [202] and Yamakawa et al., 2018 [203] for the analysis procedure.

  23. 23.

    The other case of \((-,{\tilde n})\) can be treated similarly, and the cases of (n,  +) and \((+,{\tilde n})\) are immediate from the result for a C∞v-equivariant system included in Table 14.4b in Sect. 14.4.

  24. 24.

    See, for example, Sattinger, 1983 [169] and Golubitsky, Stewart, and Schaeffer, 1988 [57].

  25. 25.

    See, for example, Sattinger, 1983 [169]; Iooss, 1986 [100]; and Golubitsky, Stewart, and Schaeffer, 1988 [57].

  26. 26.

    An alternative choice \((z_1,z_2,z_3,z_4)= (z_{1},z_{2},\overline {z_1},\overline {z_2})\) does not work, since this is not compatible with the action of \(\tilde r(\tilde \psi )\) in (16.62). Indeed, if \(z_3=\overline {z_1}\) and \(z_4=\overline {z_2}\), the action of \(\tilde r(\tilde \psi )\) on z 3 requires \(\overline {z_1} \mapsto \zeta \overline {z_1}\), which is not compatible with z 1↦ζz 1.

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Authors and Affiliations

  1. Department of Civil Engineering, Tohoku University, Sendai, Japan

    Kiyohiro Ikeda

  2. Department of Economics and Business Administration, Tokyo Metropolitan University, Hachioji, Japan

    Kazuo Murota

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Ikeda, K., Murota, K. (2019). Echelon-Mode Formation. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_16

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