Abstract
Cylindrical soils undergo complicated bifurcation behaviors due to the loss of symmetry. As a first step to model its symmetry, the dihedral group symmetry of their cross section is exploited in Chaps. 9 and 11. To exploit symmetry breaking in the axial direction, this chapter deals with a larger group D∞h \((\cong {\mathrm {O}}(2) \times {\mathbb {Z}}_{2})\), which denotes the combination of upside-down symmetry and axisymmetry of a cylindrical domain. Recursive bifurcation and mode switching are highlighted as important behaviors. The perfect system is recovered with reference to imperfect behaviors of cylindrical soils using the procedure advanced in Chap. 6. Group-theoretic bifurcation theory presented in Chap. 8 and its application to the dihedral group in Chap. 9 are foundations of this chapter. An extension to a larger symmetry group O(2) ×O(2) is to be given in Chap. 16 to detect patterns with high spatial frequencies.
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- 1.
Diverse shear bands and deformation patterns of soils and granular materials have been reported in the literature describing experimental studies; see, for example, Desrues, Lanier, and Stutz, 1985 [35]; Nakano, 1993 [140]; Melo, Umbanhowar, and Swinney, 1995 [131]; Venkataramani and Ott, 1998 [192]; Andersen et al., 2002 [2]; Wolf, König, and Triantafyllidis, 2003 [200]; and Ikeda, Sasaki, and Ichimura, 2006 [97].
- 2.
Plastic bifurcation theory (Hill and Hutchinson, 1975 [67]) was built on Hill’s theory of the uniqueness and stability of solutions of elastic–plastic solids (Hill, 1958 [66]). This theory lays a foundation of the numerical bifurcation analyses of soils developed thereafter; see, for example, Kolymbas, 1981 [115]; Vardoulakis and Sulem, 1995 [191]; and Asaoka and Noda, 1995 [6].
- 3.
In Asaoka and Nakano, 1996 [5], under constant uniform circumferential pressure (stress) σ 3, the axial strain ε a or the deviatoric stress σ a = σ 1 − σ 3 (σ 1 is the axial stress) was increased to shear the specimen. The axial strain ε a, axial stress σ 1, and volumetric strain ε v were measured to plot experimental curves, which are used for engineering-related decisions. The axial strain was obtained as the average of the shortening of the whole specimen, and the axial stress is obtained as the average on the top surface force of the specimen. A more general account of this test is given in Terzaghi and Peck, 1967 [179].
- 4.
The study in this chapter is based on Ikeda et al., 1997 [90].
- 5.
In applying these mathematical tools, the concrete form of the governing equations for the soil specimen need not be identified. It might, however, be mentioned that the Cam clay model is popular in soil mechanics (e.g., Schofield and Wroth, 1968 [171]).
- 6.
- 7.
The notations O(2) and \({\mathbb {Z}}_2\) are commonly used in the mathematical literature to denote groups isomorphic (≅) to C∞v = 〈σ v, c(φ)〉 and 〈σ h〉, respectively. To be more specific, O(2) is the two-dimensional orthogonal group consisting of 2 × 2 orthogonal matrices and \({\mathbb {Z}}_2=\{1,-1\}\) is the two-element group .
- 8.
The notation SO(2) is used in the mathematical literature to indicate a group isomorphic to C∞ = 〈c(φ)〉. To be more specific, SO(2) is the two-dimensional special orthogonal group consisting of 2 × 2 orthogonal matrices with determinant equal to one.
- 9.
- 10.
We can treat a D∞h-equivariant system similarly by setting n →∞ in an appropriate way.
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Ikeda, K., Murota, K. (2019). Bifurcation Behaviors of Cylindrical Soils. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_14
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