The formation of tabular compaction-band arrays: Theoretical and numerical analysis

The bifurcation analysis of compaction banding is extended to the formation of a tabular discrete compaction-band array. This analysis, taken together with the results of finite-difference simulations, shows that the bifurcation results in the formation of intermittent loading (elastic-plastic) and unloading (elastic) bands. The obtained analytical solution relates the spacing parameter χ (the ratio between the band thickness to the band-to-band distance) to all constitutive and stress-state parameters. Both this solution and numerical models reveal strong dependence of χ on the hardening modulus h: χ increases with h reduction. The band thickness in the numerical models is mesh dependent, but in terms of mesh-zone-size varies only from ∼2 to 4 depending on the constitutive parameters and independently on the mesh resolution. The thickness of the “elementary” compaction bands in real granular materials is equal to a few grain sizes. It follows that one grid zone in the numerical models corresponds approximately to one grain in the real material. The numerical models reproduce both discrete and continuous propagating compaction banding observed in the rock samples. These phenomena were shown to be dependent on the evolution of h and the dilatancy factor with deformation.