A Variational Gradient Plasticity Model allowing for Shear Band Localization

Plasticity in nonlinear solid mechanics describes the development of inelastic deformations in engineering materials when a critical stress threshold is exceeded. Experimental observations at the microstructural scale have demonstrated that plastic behaviour depends on grain size, a phenomenon that classical plasticity theories fail to capture due to the absence of an intrinsic material length scale [1]. To address this limitation, we introduce a variational gradient plasticity model incorporating non- local first-order terms, specifically the curl of the plastic strain field, which provides a homogenized measure of dislocation distributions and internal lattice incompatibilities. Unlike traditional gradient plasticity models that employ quadratic regularization terms to smooth plastic strain gradients, our approach replaces this with a sub-quadratic term that introduces threshold effects, fundamentally altering the plastic response (see [2] and [3]). This novel formulation influences the plastic threshold and enhances the representation of shear band formation by enabling sharper localization effects rather than diffusive smoothing. The impact of these modifications is analyzed through a two-dimensional shear problem involving a hollow cylinder, where both analytical and numerical solutions illustrate the role of first- and second-order curl terms in plastic evolution. By emphasizing dislocation interactions and their role in plastic localization, this study provides a new framework for modeling size-dependent plasticity with greater fidelity. The proposed approach offers fresh insights into dislocation nucleation, defect formation, and the evolution of curved plastic zones, paving the way for more accurate descriptions of material behavior under extreme conditions.