Eshelby's solution for an ellipsoidal inhomogeneous inclusion in an infinite elastic body is applied to compaction and shear-enhanced compaction bands in the Aztec sandstone at Valley of Fire State Park, NV. The inclusion and matrix are linear elastic and isotropic, but have different elastic moduli, and a remote stress represents tectonic loading. A prescribed uniform strain within the inclusion accounts for inelastic compaction for a porosity change from 25 to 10%. Differences in elastic moduli between the matrix and inclusion are based on laboratory data. We generalize earlier results, limited to 2D and axisymmetric geometries, by considering ellipsoids with different intermediate and greatest axial lengths, consistent with field observations. Stiffness contrasts and non-circular tip-line shapes produce modest concentrations of the remote stress, but compaction strains of 1–10% produce significant triaxial compressive stress concentrations, which presumably are responsible for band propagation. The plastic strain is triaxial, but dominated by the normal strain across the inclusion. The stress diminution on the band flank is easily overcome by minor increases in the tectonic loading, enabling bands to be closely spaced. For the shear-enhanced band, if the plastic shear and normal strains are approximately equal, the ratio of shear to normal stress is about 1.3 at the tip.