Image simulation of uniform materials subjected to recursive bifurcation

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 mat erials is proposed here by highlighting recursive symmetry-breaking bifurca tion 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 con ditions, we employ the infinite-periodic-domain approximation which assumes that the domain is periodically extended in the x- and y-directions, respe ctively. Since real material properties manifest itself sufficiently away f rom the boundaries and usually form some characteristic patterns, the use o f periodic boundaries is essential in the simulation of true material prope rties. Rules of the recursive bifurcation, which are expressed in terms of a hierarchy of subgroups labeling the symmetries of deformation patterns, a re constructed by extending the pre-existing group-theoretic studies for th is domain. The use of periodic boundaries has led to the emergence of the s ubgroups labeling stripe and echelon symmetries that disappear if these bou ndaries are not used. These rules of bifurcation are interpreted in terms o f the double Fourier series to prepare for the image analysis of deformatio ns in a rectangular domain. The use of the Fourier series has physical nece ssity in that the direct bifurcation modes of uniform domains are always ha rmonic and that periodic properties are better expressed in the frequency d omain. Mode interference with high frequencies after bifurcation is advance d as the mechanism of localization of deformations. The computational analy sis on a rectangular domain (plate) with periodic boundaries at four sides is conducted to present a numerical example of echelon-mode formation throu gh recursive (cascade) bifurcation. The procedure for image simulation is a pplied to a few uniform materials, including: kaolin and steel specimens. T he intensity of the digital images of the deformation patterns of these spe cimens in the frequency domain is successfully classified with the use of t he rules of recursive bifurcation. As a result of these. the transient proc ess of deformations, which was not discernible by the mere visual observati ons and was less understood so far. is identified based on a firm theoretic al basis. The recursive bifurcation has thus been acknowledged to be the un derlying mechanism of pattern formation of uniform materials. (C) 2001 Else vier Science Ltd. All rights reserved.