Deformations of uniform materials are well known to display characteristic geometrical patterns such as en echelon cracks. A systematic procedure for the image simulation of the progress of deformation patterns of uniform materials is proposed here by highlighting recursive symmetry-breaking bifurcation as the fundamental mechanism to generate patterns. We here focus on a rectangular domain with periodic boundaries. That is, to better express the local uniformity at the sacrifice of the consistency with the boundary conditions, we employ the infinite-periodic-domain approximation which assumes that the domain is periodically extended in the x- and y-directions, respectively. Since real material properties manifest itself sufficiently away from the boundaries and usually form some characteristic patterns, the use of periodic boundaries is essential in the simulation of true material properties. Rules of the recursive bifurcation, which are expressed in terms of a hierarchy of subgroups labeling the symmetries of deformation patterns, are constructed by extending the pre-existing group-theoretic studies for this domain. The use of periodic boundaries has led to the emergence of the subgroups labeling stripe and echelon symmetries that disappear if these boundaries are not used. These rules of bifurcation are interpreted in terms of the double Fourier series to prepare for the image analysis of deformations in a rectangular domain. The use of the Fourier series has physical necessity in that the direct bifurcation modes of uniform domains are always harmonic and that periodic properties are better expressed in the frequency domain. Mode interference with high frequencies after bifurcation is advanced as the mechanism of localization of deformations. The computational analysis on a rectangular domain (plate) with periodic boundaries at four sides is conducted to present a numerical example of echelon-mode formation through recursive (cascade) bifurcation. The procedure for image simulation is applied to a few uniform materials, including: kaolin and steel specimens. The intensity of the digital images of the deformation patterns of these specimens in the frequency domain is successfully classified with the use of the rules of recursive bifurcation. As a result of these, the transient process of deformations, which was not discernible by the mere visual observations and was less understood so far, is identified based on a firm theoretical basis. The recursive bifurcation has thus been acknowledged to be the underlying mechanism of pattern formation of uniform materials.
- Group-theoreic bifurcation theory
- Image simulation
- Recursive bifurcation
ASJC Scopus subject areas