We employ a novel discrete element method (DEM) force formulation to simulate adhesive wear and assess the effects of material and loading parameters on the properties of the third-body layer (TBL) formed during sliding motion. The study emphasizes the role of a material's critical length scale d^* in the rheology of the TBL. This critical length scale is already known for controlling the size of smallest wear particles. We observe the emergence of a several wear regimes involving wear particle creation and aggregation, with limited effect from d^* on TBL properties. Instead, material strength and surface energy have a profound influence. This study opens up new avenues for exploration of larger systems, three-dimensional setups, and other loading conditions.