Abstract
Bifurcation mechanism of honeycomb structures is elucidated by the study of a that is the direct product of O(2) and two reflection group. A flower pattern is theoretically assessed to branch from a triple bifurcation point and is actually found by a numerical analysis of a honeycomb cellular solid. Other bifurcating patterns of interest are found in this study through the analysis of bifurcation points with the multiplicity of six and twelve. Fundamentals of group representation theory in Chap. 7 and group-theoretic bifurcation theory in Chap. 8 are foundations of this chapter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Characteristic deformation patterns of honeycomb structures subjected to uniaxial and biaxial in-plane compression were found during experiments (e.g., Gibson and Ashby, 1997 [53]). In particular, a flower mode was observed experimentally (e.g., Papka and Kyriakides, 1999 [153]) and was simulated successfully by finite-element analyses (e.g., Guo and Gibson, 1999 [59]).
- 2.
The theoretical and numerical analyses in this chapter are based on Saiki, Ikeda, and Murota, 2005 [166]. Corrections and revisions were made in the second edition of this book to supplement deficiencies and to present more details. In this third edition, a further extension is made based on Ikeda, Murota, and Akamatsu, 2012 [85]; Ikeda and Murota, 2014 [84]; Ikeda et al., 2014 [86]; and Ikeda, Murota, and Takayama, 2017 [89].
- 3.
- 4.
These patterns, for example, are hexagons, antihexagons, rolls, regular triangle, and patchwork quilt. The hexagonal lattice is employed in the description of convection of fluids and nematic liquid crystals (cf., Peacock et al., 1999 [154]; Golubitsky and Stewart, 2002 [56]; and Chillingworth and Golubitsky, 2003 [23]). For related issues, see Buzano and Golubitsky, 1983 [19]; Melbourne, 1999 [130]; and Bressloff et al., 2001 [17]. Refer to Crawford, 1994 [30] for a study of a square lattice.
- 5.
Although \(\mathbb {Z}_{n}\) is isomorphic to Cn, it is appropriate to use \(\mathbb {Z}_{n}\) in the context of this chapter.
- 6.
Notation \(\ \dot {+} \ \) for a semidirect product, instead of \(\ltimes \), is used, e.g., in Golubitsky, Stewart, and Schaeffer, 1988 [57].
- 7.
- 8.
Details of this analysis are available in Saiki, Ikeda, and Murota, 2005 [166].
- 9.
Details of these irreducible representations can be found in Ikeda and Murota, 2014 [84, Chapter 6].
- 10.
Group-theoretic bifurcation analysis of six-dimensional irreducible representations of the group \({\mathrm {D}}_6 \ltimes {\mathrm {T}}^2\), where T2 means a two-dimensional torus, was conducted to show the existence of possible bifurcating patterns: hexagons, rolls, and triangles (Kirchgässner, 1979 [112]; Buzano and Golubitsky, 1983 [19]; Dionne and Golubitsky, 1992 [38]; Golubitsky and Stewart, 2002 [56]).
- 11.
The condition (17.89) plays no critical role in solving the bifurcation equations although it puts some constraints on some terms.
- 12.
Group-theoretic bifurcation analysis of 12-dimensional irreducible representations of the group \({\mathrm {D}}_6 \ltimes {\mathrm {T}}^2\), where T2 means a two-dimensional torus, was conducted to show the existence of possible bifurcating patterns: simple hexagons and super hexagons (Kirchgässner, 1979 [112]; Dionne, Silber, and Skeldon, 1997 [39]; Judd and Silber, 2000 [104]).
- 13.
References
Bénard, H. (1900) Les tourbillons cellulaires dans une nappe liquide. Rev. Gén. Sci. Pure Appl. 11, 1261–1271, 1309–1328.
Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J. (2001) Scalar and pseudoscalar bifurcations: Pattern formation in the visual cortex. Nonlinearity 14 (4), 739–775.
Buzano, E., Golubitsky, M. (1983) Bifurcation on the hexagonal lattice and the planar Bénard problem. Phil. Trans. Roy. Soc. London A 308, 617–667.
Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Clarendon Press, London.
Chillingworth, D.R.J., Golubitsky, M. (2003) Planar pattern formation in bifurcation from a homeotropic nematic liquid crystal. In: Bifurcation, Symmetry and Patterns, ed. J. Buescu et al., Trends in Mathematics. Birkhäuser, Basel.
Crawford, J.D. (1994) \(D_4\dot {+}T^2\) mode interactions and hidden rotational symmetry. Nonlinearity 7 (3), 697–739.
Dionne, B., Golubitsky, M. (1992) Planforms in two and three dimensions. Zeit. angew. Math. Phys. 43 (1), 36–62.
Dionne, B., Silber, M. and Skeldon, A.C. (1997) Stability results for steady, spatially, periodic planforms. Nonlinearity 10, 321–353.
Gibson, L.J., Ashby, M.F. (1997) Cellular Solids: Structure and Properties, 2nd ed. Cambridge University Press, Cambridge, UK.
Golubitsky, M., Stewart, I. (2002) The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Progress in Mathematics, Vol. 200. Birkhäuser, Basel.
Golubitsky, M., Stewart, I., Schaeffer, D.G. (1988) Singularities and Groups in Bifurcation Theory, Vol. 2. Applied Mathematical Sciences, Vol. 69. Springer-Verlag, New York.
Guo, X.E., Gibson, L.J. (1999) Behavior of intact and damaged honeycomb: a finite element study. Internat. J. Mech. Sci. 41 (1), 85–105.
Ikeda, K., Aizawa, H., Kogure, Y., Takayama, Y. (2018) Stability of bifurcating patterns of spatial economy models on a hexagonal lattice. Internat. J. Bifurcation Chaos 28 (11), 1850138 (30 pages).
Ikeda, K., Murota, K. (2014) Bifurcation Theory for Hexagonal Agglomeration in Economic Geography. Springer-Verlag, Tokyo.
Ikeda, K., Murota, K., Akamatsu, T. (2012) Self-organization of Lösch’s hexagons in economic agglomeration for core–periphery models. Internat. J. Bifurcation Chaos 22 (8), 1230026 (29 pages).
Ikeda, K., Murota, K., Akamatsu, T., Kono, T., Takayama, Y. (2014) Self-organization of hexagonal agglomeration patterns in new economic geography models. J. Economic Behavior & Organization 99, 32–52.
Ikeda, K., Murota, K., Takayama, Y. (2017) Stable economic agglomeration patterns in two dimensions: Beyond the scope of central place theory. J. Regional Science 57 (1), 132–172.
Judd, S.L., Silber, M. (2000) Simple and superlattice turning patterns in reaction-diffusion systems: bifurcation, bistability, parameter collapse. Physica D 136 (1–2), 45–65.
Kirchgässner, K. (1979) Exotische Lösungen Bénardschen Problems, Math. Meth. Appl. Sci. 1, 453–467.
Koschmieder, E.L. (1966) On convection of a uniformly heated plane. Beitr. zur Phys. Atmosphäre 39, 1–11.
Koschmieder, E. L. (1993) Bénard Cells and Taylor Vortices. Cambridge Monograph on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge, UK.
Kudrolli, A., Pier, B., Gollub, J.P. (1998) Superlattice patterns in surface waves. Physica D 123 (1–4), 99–111.
Melbourne, I. (1999) Steady-state bifurcation with Euclidean symmetry. Trans. Amer. Math. Soc. 351 (4), 1575–1603.
Ohno, N., Okumura, D., Noguchi, H. (2002) Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation. J. Mech. Phys. Solids 50 (5), 1125–1153.
Papka, S.D., Kyriakides, S. (1999) Biaxial crushing of honeycombs—Part I: Experiments. Internat. J. Solids Structures 36 (29), 4367–4396.
Peacock, T., Mullin, T., Binks, D. J. (1999) Bifurcation phenomena in flows of a nematic liquid crystal. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 9 (2), 427–441.
Saiki, I., Ikeda, K., Murota, K. (2005) Flower patterns appearing on a honeycomb structure and their bifurcation mechanism. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 15 (2), 497–515.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ikeda, K., Murota, K. (2019). Flower Patterns on Honeycomb Structures. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-21473-9_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21472-2
Online ISBN: 978-3-030-21473-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)