| The onset and evolution of tabular compaction bands is studied based on the discontinuous
bifurcation analysis and finite-difference simulations. In numerical models, the bands are
initiated as constitutive instabilities resulting from the deformation bifurcation. Band spacing,
length and aspect strongly depend on the constitutive parameters and particularly on the
hardening modulus h, both spacing and length rapidly increasing with h. Compaction
banding is only possible when h lies within certain limits hmin and hmax defined by other
parameters. If h > hmax, the deformation localisation is either impossible at all or
occurs in the form of shear banding, depending on the parameters. The transition from shear
to compaction banding is gradual and corresponds to the formation of crooked or zigzag
bands similar to those obtained in the experimental rock tests and also observed in the field.
These bands are very dense and have small wavelength when h approaches hmin. On the
other hand, when h approaches hmax, the forming compaction bands are linear and
long. After the initiation of these bands (from deformation bifurcation) some of
them are dying, while others continue a post-bifurcation evolution accumulating
the inelastic deformation/damage and compressive stress at their tips. The stress
concentration/increase, however, does not exceed 0.1% of the background value. Starting
from some stage, the bands begin to propagate similarly to cracks. The propagation then
slows down simultaneously with the beginning of bands’ thickening that occurs due to
the incorporation of not yet compacted material at the band flanks. The response
of the already compacted “core” part of the band becomes mostly elastic. Then
the propagation practically stops and the bands undergo only the heterogeneous
thickening, maximal in the middle of the band and reducing toward its tips. This scenario
obtained directly in the numerical models (without any specific hypotheses about the
propagation mechanism) appears more complicated than what can be expected from
the LEFM anti-crack model. The band propagation distance is proportional to the
initial (resulted from the bifurcation) band length that in turn is proportional to the
hardening modulus and theoretically can reach infinity. These bands are thicker in the
central band segment and are progressively thinning toward the ends, while the
thickness of dense zigzag bands is rather uniform. The microphysics of the observed
difference between the bands of various types is discussed and related to the evolution
(continuous versus discontinuous) of the hardening modulus with inelastic deformation. |