Carbide Nanoparticle Hardness (2D): Dislocation Pile-Up & Hall-Petch

Metallic and covalent carbide nanoparticles — WC, TiC, SiC — get harder as their crystalline grains shrink, and this 2D companion simulator derives that trend from its actual microscopic cause instead of only stating the empirical curve. A slip plane crosses a real 2D grain-boundary tessellation, and gliding dislocations pile up against the nearest boundary; classical pile-up theory (Eshelby–Frank–Nabarro) predicts both how many dislocations fit in that pile-up and the shear stress needed to keep pushing more in, and solving the two together for the applied stress reproduces the 1/√d Hall–Petch scaling directly from first principles — smaller grains support shorter pile-ups but demand higher stress to transmit slip across the boundary. Alongside that mechanism, the same macroscopic Hall–Petch / inverse Hall–Petch hardness curve used for real WC–Co, TiC and SiC data is plotted live with the current grain size marked, and a true-geometry Vickers indent cross-section — complete with a binder-driven pile-up or sink-in bulge at its edges, the well documented behaviour real nanoindentation traces show — presses into the material under an adjustable load.