Flower patterns appearing on a honeycomb structure and their bifurcation mechanism

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 15, Issue 2, 2005, Pages 497-515

  Saiki, Isao ^a   Ikeda, Kiyohiro ^b   Murota, Kazuo ^c,d  

^a UTSUNOMIYA UNIVERSITY   (Japan)

^b TOHOKU UNIVERSITY   (Japan)

^c UNIVERSITY OF TOKYO   (Japan)

^d JAPAN SCIENCE AND TECHNOLOGY AGENCY   (Japan)

Author keywords

Bifurcation mechanism; Flower pattern; Group theoretic bifurcation theory; Honeycomb structure

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY; FLOW OF FLUIDS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; PROBLEM SOLVING;

BIFURCATION MECHANISM; FLOWER PATTERN; GROUP-THEORETIC BIFURCATION THEORY; IN-PLANE COMPRESSION;

HONEYCOMB STRUCTURES;

EID: 18644372396     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812740501217X     Document Type: Article

References (42)

1. Bénard, H. [1900] "Les tourbillons cellulaires dans une nappe liquide," Rev. Gén. Sci. Pure Appl. 11, 1261-1271, 1309-1328.
2. Bressloff, P. C., Cowan, J. D., Golubitsky. M. & Thomas, P. J. [2001] "Scalar and pseudoscalar bifurcations: Pattern formation in the visual cortex," Nonlinearity 14, 739-775.
3. Buzano, E. & Golubitsky, M. [1983] "Bifurcation on the hexagonal lattice and the planar Bénard problem," Phil. Trans. Roy. Soc. London A308. 617-667.
4. Chung, J. & Waas, A. M. [1999] "Compressive response and failure of circular cell polycarbonate honeycombs under inplane uniaxial stresses," ASME J. Eng. Mater. Tech. 121, 494-502.
5. Chung, J. & Waas, A. M. [2001] "In-plane biaxial crush response of polycarbonate honeycombs," J. Eng. Mech. 127, 180-193.
6. 2 mode interactions and hidden rotational symmetry," Nonlinearity 7, 697-739.
7. 4-symmetric maps with hidden Euclidean symmetry," in Pattern Formation: Symmetry Methods and Applications, eds. Chadam, J., Golubitsky, M., Langford, W. & Wetton, B. (American Math. Soc.), pp. 93-124.
8. Dionne, B. & Golubitsky, M. [1992] "Planforms in two and three dimensions," Zeit. Angew. Math. Phys. 43, 36-62.
9. Gibson, L. J., Ashby, M. F., Zhang, J. & Triantafillou, T. C. [1989] "Failure surfaces for cellular materials under multiaxial loads - I. Modeling," Int. J. Mech. Sci, 31, 635-663.
10. Gibson, L. J. & Ashby, M. F. [1997] Cellular Solids: Structure and Properties, 2nd Ed. (Cambridge University Press).
11. Golubitsky, M., Stewart, I. & Schaeffer, D. G. [1988] Singularities and Groups in Bifurcation Theory, Vol. 2 (Springer).
12. Golubitsky, M. & Stewart, I. [2002] The Symmetry Perspective (Birkhäuser Verlag).
13. Golubitsky, M. & Chillingworth, D. R. J. [2003] "Bifurcation and planar pattern formation for a liquid crystal," in Bifurcation, Symmetry and Patterns, ed. Buescu, J. (Birkhäuser), pp. 55-66.
14. Guo, X. E. & Gibson, L. J. [1999] "Behavior of intact and damaged honeycomb: A finite element study," Int. J. Mech, Sci. 41, 85-105.
15. Ikeda, K., Murota, K. & Fujii, H. [1991] "Bifurcation hierarchy of symmetric structures," Int. J. Solids Struct. 27, 1551-1573.
16. Ikeda, K., Murota, K. & Nakano, M. [1994] "Echelon modes in uniform materials," Int. J. Solids Struct. 31, 2709-2733.
17. Ikeda, K., Murakami, S., Saiki, I., Sano, I. & Oguma, N. [2001] "Image simulation of uniform materials subjected to recursive bifurcation," Int. J. Eng. Sci. 39, 1963-1999.
18. Ikeda, K. & Murota, K. [2002] Imperfect Bifurcation in Structures and Materials - Engineering Use of Group-Theoretic Bifurcation Theory (Springer).
19. Iwakuma, T., Hasegawa, A., Nishino, F. & Kuranishi, S. [1987] "Principle and numerical check of a stiffness equation for plane frames," Proc. JSCE, Structural Eng./Earthquake Eng. 4, 73s-83s.
20. Judd, S. L. & Silber, M. [2000] "Simple and superlattice turning patterns in reaction-diffusion systems: Bifurcation, bistability, and parameter collapse," Physica D136, 45-65.
21. Kirchgässner, K. [1979] "Exotische lösungen Bénard-Schen problems," Math. Meth. Appl. Sci. 1, 453-467.
22. Klintworth, J. W. & Stronge, W. J. [1988] "Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs," Int. J. Mech. Sci. 30, 273-293.
23. Koschmieder, E. L. [1966] "On convection of a uniformly heated plane," Beiträge zur Physik der Atmosphäre 39, 1-11.
24. Koschmieder, E. L. [1993] Bénard Cells and Taylor Vortices (Cambridge).
25. Kudrolli, A., Pier, B. & Gollub, J. P. [1998] "Superlattice patterns in surface waves," Physica D123, 99-111.
26. Melbourne, I. [1999] "Steady-state bifurcation with Euclidean symmetry," Trans. Amer. Math. Soc. 351, 1575-1603.
27. Murota, K., Ikeda, K. & Terada, K. [1999] "Bifurcation mechanism underlying echelon-mode formation," Comput. Meth. Appl. Mech. Eng. 170, 423-448.
28. Oguni, K., Hori, M. & Ikeda, K. [1997] "Analysis on evolution pattern of periodically distributed defects," Int. J. Solids Struct. 34, 3259-3272.
29. 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, 1125-1153.
30. Okumura, D., Ohno, N. & Noguchi, H. [2002] "Post-buckling analysis of elastic honeycombs subject to inplane biaxial compression," Int. J. Solids Struct. 39, 3487-3503.
31. Okumura, D., Ohno, N. & Noguchi, H. [2004] "Elasto-plastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids," J. Mech. Phys. Solids 52, 641-666.
32. Ooue, K., Saiki, I., Terada, K. & Nakajima, A. [2004] "Nonlinear multi-scale modeling with frame elements for cellular materials," Struct. Eng./Earthquake Eng. 21, 67s-75s.
33. Papka, S. D. & Kyriakides, S. [1994] "In-plane compressive response and crushing of honeycomb," J. Mech. Phys. Solids 42, 1499-1532.
34. Papka, S. D. & Kyriakides, S. [1999a] "Biaxial crushing of honeycombs - Part I: Experiments," Int. J. Solids Struct. 36, 4367-4396.
35. Papka, S. D. & Kyriakides, S. [1999b] "Biaxial crushing of honeycombs - Part II: Analysis," Int. J. Solids Struct. 36, 4397-4423.
36. Peacock, T., Mullin, T. & Binks, D. J. [1999] "Bifurcation phenomena in flows of a nematic liquid crystal," Int. J. Bifurcation and Chaos 9, 427-441.
37. Saiki, I., Terada, K., Ikeda, K. & Hori, M. [2002] "Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling," Comput. Meth. Appl. Mech. Eng. 191, 2561-2585.
38. Sattinger, D. H. [1978] "Group representation theory, bifurcation theory and pattern formation," J. Funct. Anal 28, 58-101.
39. Sattinger, D. H. [1979] Group Theoretic Methods in Bifurcation Theory, Lecture Notes in Mathematics, Vol. 762 (Springer-Verlag).
40. Tanaka, R., Saiki, I. & Ikeda, K. [2002] "Group-theoretic bifurcation mechanism for pattern formation in three-dimensional uniform materials," Int. J. Bifurcation and Chaos 12, 2767-2797.
41. Triantafyllidis, N. & Schraad, M. W. [1998] "Onset of failure in aluminum honeycombs under general inplane loading," J. Mech. Phys. Solids 46, 1089-1124.
42. Zhu, H. X. & Mills, N. J. [2000] "The inplane non-linear compression of regular honeycombs," Int. J. Solids Struct. 37, 1931-1949.