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Honeycomb structures under compression display illuminative geometrical patterns. As an example, Fig. 16.1(a) shows the so-called flower mode; a flowerlike pattern in (b), which is cut out from (a), comprises a regular hexagon and six identical cells surrounding this hexagon. Presented in (c) are its variants with different symmetries. In the numerical bifurcation analysis of the honeycomb structure to search for new patterns, it is pertinent to take advantage of group-theoretic analytical information. [ABSTRACT FROM AUTHOR]

The deformation pattern change of cylindrical soil specimens at the later stage has been discussed in Chapter 13. In this chapter, we move on to investigate pattern formation at the earlier stage with high spatial frequencies. [ABSTRACT FROM AUTHOR]

The shear band formation of a steel specimen subjected to tension has been ascribed to the direct bifurcation from a uniform state in plastic bifurcation theory, and is simulated by numerical analyses. However, direct bifurcation alone cannot account for the complexity of experimental deformation patterns. The recursive bifurcation that entails complex deformation patterns can be advanced as an underlying mechanism to create such complexity, as described in Chapters 13 and 14. [ABSTRACT FROM AUTHOR]

It was clarified in Chapter 5, for simple critical points, that the probabilistic properties of critical loads can be formulated in an asymptotic sense (when imperfections are small). In this chapter, this formulation is extended to a Dn-equivariant system that potentially has simple and double bifurcation points. For a simple critical point of a Dn-equivariant system, which is either a limit point or a pitchfork bifurcation point (cf., §8.3.1), the relevant results presented in Chapter 5are applicable. [ABSTRACT FROM AUTHOR]

Triaxial compression tests of soils and sands are conducted to ascertain their mechanical properties in soil mechanics, as shown in Fig. 13.1. A cylindrical soil or sand specimen under compression exhibits the concentration of strain into a narrow damaged zone, which is called a shear band. Under constant uniform circumferential pressure (stress) σ3, the axial strain εa or the deviatoric stress σa = σ1 -σ3 ( σ1 is the axial pressure) is increased to shear the specimen. [ABSTRACT FROM AUTHOR]

In the preceding chapters in Part II, several procedures to tackle the perfect and imperfect behaviors of a Dn-equivariant system were introduced and applied to simple examples. Each procedure, however, can throw light on only a fraction of all the bifurcation behaviors, especially in experiments. It might be in order, in this chapter, to investigate the perfect and imperfect behaviors of realistic systems, by a synthetic application of those procedures, to demonstrate their capability. [ABSTRACT FROM AUTHOR]

Group representation theory guarantees that the Jacobian matrix of symmetric systems can be transformed to a block-diagonal form (cf., §7.7.1). This chapter presents an efficient computational method for this block-diagonalization. [ABSTRACT FROM AUTHOR]

The critical load of a structural system undergoing bifurcation is sensitive to imperfections (cf.,Chapter 3). In view of the engineering demand to consider the imperfection that reduces the critical load most rapidly, the worst imperfection1 has been formulated in Chapter 4 for simple critical points. [ABSTRACT FROM AUTHOR]

Group-theoretic bifurcation theory has been presented in the preceding chapter. In this chapter, this theory is applied to the analysis of a system that is equivariant to the simplest group, the dihedral group Dn, which represents the symmetry of the regular n-sided polygon. [ABSTRACT FROM AUTHOR]

Extensive studies of structures and materials undergoing perfect and imperfect bifurcation have been conducted. Nevertheless, a gap separating mathematical theory and engineering practice remains in the experimentation of materials undergoing bifurcation. Such a gap may be ascribed to the following three essential difficulties. [ABSTRACT FROM AUTHOR]

Qualitative aspects of symmetry-breaking bifurcation can be described by grouptheoretic bifurcation theory. In view of the symmetry of the system under consideration, possible critical points and bifurcated solutions can be classified, and the behavior of these solutions in a neighborhood of each critical point can be investigated thoroughly by the Liapunov–Schmidt reduction. An extremely important finding of this theory is that the mechanism of such bifurcation does not depend on individual material or physical properties but on the symmetry of the system under consideration. [ABSTRACT FROM AUTHOR]

Even small imperfections can sharply reduce the strength of structures, such as shells, undergoing bifurcation. Classical linearized theory was inadequate to explain such sharp reductions for shells. This inadequacy was resolved through the implementation of nonlinearity and imperfections into the governing equation. [ABSTRACT FROM AUTHOR]

Bifurcation, which means the emergence of multiple solutions for the same value of parameter f, is induced by the criticality of the Jacobian matrix of the system, as demonstrated using examples in the previous chapter (cf., §1.2.2). The ˵bifurcation equation″ is a standard means to describe bifurcation behavior. In a neighborhood of a simple critical point, for example, a set of equilibrium equations is reduced to a single bifurcation equation, by condensing the influence of a number of independent variables into a single scalar variable by the implicit function theorem. [ABSTRACT FROM AUTHOR]

This book offers several systematic methods based on up-to-date mathematics to untangle the mechanism of bifurcation of structures and materials, such as soil, sand, kaolin, and concrete. Throughout this book, we place more emphasis on engineering pragmatism than on mathematical rigor. We offer, in this first chapter, an overview of the book using a series of illustrative examples and photographs. [ABSTRACT FROM AUTHOR]