Conditions for compaction and shear bands in a transversely isotropic material

This paper derives conditions for localized deformation for a transversely isotropic constitutive relation intended to model the response of geological materials in the axisymmetric compression test. The analysis considers the possibility of shear bands, with dilation or compaction, and pure compaction bands. The latter are planar zones of localized pure compressive deformation (without shear) that form perpendicular to the direction of the maximum principal compressive stress. Compaction bands have been observed in porous rock in the field and in the laboratory. They are predicted to occur when the incremental tangent modulus for uniaxial deformation vanishes. The critical value of the tangent modulus E for constant lateral stress is -9Kvr/2, where v is the negative of the ratio of increments of lateral to axial deformation (at constant lateral stress), r is the ratio of axial to lateral stress increments causing zero axial deformation, and K is the modulus relating increments of lateral stress and deformation. The expression for the critical tangent modulus for shear band formation is more complex and depends, in addition to r, v, and K, on the shear moduli G₁ and G_t, governing increments of shear in planes parallel and perpendicular to the axis of symmetry, respectively. Uncertainty about material parameters prevents a detailed comparison with observations but the results are consistent with observations of low angle shear bands (with normals less than 45° from the symmetry axis) for compressive volumetric strain (v < 1/2). In addition, the critical tangent modulus for such bands may be positive if G₁ and G_t are small relative to K and r is around unity.