In this research, the improvement in sheet metal formability obtained through the utilization of a mineral grease in solid oil type (SMG) pressure medium in the hydrodynamic deep drawing process assisted by radial pressure (HDDRP) was investigated by experiments and finite element method (FEM) simulations on flat‐bottom cups realized with pure copper and DIN 1.0338 steel. To determine the process parameters ranges, a FEM model was developed in ABAQUS/Explicit and validated against experimental results. Afterward, experiments were carried out using two pressure mediums with different viscosities, id est water, and SMG. Considering the results relevant for both sheet materials, the SMG medium allowed an average reduction of the sheet metal thinning of 15% and 10% on the corner edges while also granting a negligible thickness variation on the cup walls. On the other hand, the higher viscosity of the SMG, in comparison to the water medium, also resulted in an average 16% increase in the forming force when compared to the water pressure medium.This article is protected by copyright. All rights reserved.