Nucleation in rank-one gradient plasticity. Exact solutions and geometry-dependent regimes

We investigate plastic nucleation in solids within a strain-gradient plasticity framework featuring a rank-one defect energy, formulated as an incremental convex variational problem. The designation rank-one refers to the fact that the defect energy is positively one-homogeneous with respect to the curl of the plastic strain. The core feature of the framework is the variational selection of the plastic support, which regularizes the singular localization of classical plasticity into a distributed nucleus of finite width and introduces a genuine strengthening effect: the yield threshold is elevated by a discrete energetic barrier, while the post-yield response remains perfectly plastic. For an annular domain under azimuthal shear, we identify two qualitatively distinct nucleation regimes governed by the interplay between the internal length scale and the domain geometry. In the first regime, plasticity spreads over the entire domain in a curl-free compatible configuration. In the second, it localizes in an inner nucleus sealed by a concentrated geometric necessary dislocations wall at a locked front that does not advance upon further loading. Closed-form analytical solutions are derived for both regimes without any a priori assumption on the plastic support, providing exact benchmarks for the numerical simulations. In the localized regime, both the plastic zone size and the yield threshold follow a square-root scaling law in the internal length scale, in contrast to the linear scaling of previous rank-one models where the plastic support is prescribed by microstructural constraints. Numerical simulations on elliptical geometries show that the scaling exponents and the nucleation pattern depend sensitively on the domain shape, identifying geometry as a control variable of the nucleation process on a par with the internal length scale.