Echelon modes in uniform materials

The mechanism of the "echelon" mode-a series of parallel short wrinkles of soils that looks like a flight of stairs or wild geese arranged in formation-observed in the triaxial compression test is revealed by means of the group-theoretic bifurcation theory. The study of the direct bifurcation of uniform materials in cylindrical domains with periodic boundaries has revealed the existence of various kinds of deformation patterns. In particular, a classical diamond (checkerboard) pattern solution and a pair of "oblique" stripe pattern solutions simultaneously branch at a bifurcation point of multiplicity four. With the use of the periodic boundaries these oblique stripe solutions, which cannot be directly obtained by the customary bifurcation analysis with non-periodic boundaries, have thus been made accessible. The secondary bifurcation from these stripe solutions realizes the "echelon" mode. The triaxial compression tests on soil specimens have demonstrated that the pattern of wrinkles, shear bands and echelon modes of the specimens do indeed follow the present theory, and can be understood as a natural consequence of pattern selection, or, in other words, symmetry-breaking bifurcation.