Nonlocal implicit gradient enhancements for strain localization informed by controllability criteria for plastic solids

Nonlocal implicit gradient enhancements are widely used to suppress mesh dependency in simulations involving strain localization. For example, nonlocality is often introduced through internal variables that account for possible material softening. This work, however, shows that this approach may become ineffective if the solid displays plastic non-normality (i.e., non-associated plastic flow). For this purpose, we consider an over-nonlocal formulation and mathematically inspect the conditions at which regularization is lost in the presence of plastic non-normality. Specifically, such loss of regularization is linked to the loss of uniqueness and/or existence of the incremental plastic response that is kinematically compatible with the development of a deformation band. By doing so, we find a lower limit for the admissibility of the parameters controlling the effectiveness of nonlocal implicit gradient regularization. Furthermore, we show that such a lower limit is regulated by a plastic modulus reflecting the loss of controllability of the constitutive response, and, hence, depends on the degree of plastic non-normality. We also derive a closed-form expression relating the thickness of the deformation band to both the controllability modulus and gradient regularization constants, which suggests that the thickness of the process zone may change in response to the prevailing plastic flow characteristics and evolve during active plastic deformation. The proposed nonlocal enhancement is applied to a non-associated elasto-plastic model for porous sedimentary rocks, which is capable of displaying both shear-dominated and compaction-dominated bands. Numerical simulations reveal that effective regularization can be enforced only when the over-nonlocal weighting coefficient is larger than the above-mentioned lower limit.