Contact dynamics in ball mills with ellipsoidal, spherical and cylindrical shapes

High-energy ball milling is a key method in materials processing, though the mechanics of collisions between balls and jar walls are not fully understood. This work presents an analytical model, based on Hertzian contact theory, to describe impacts in four cases: a sphere against a prolate ellipsoidal wall, an oblate ellipsoidal wall, another sphere, and a cylindrical wall. The model provides formulas for maximum force, pressure, deformation, and contact area, depending on material, geometry, and speed. Tests with steel balls at 5.0 m/s show that larger balls generate higher forces but lower pressures, due to larger contact areas. Prolate and oblate walls show opposite angular effects, while the cylindrical case is the least conforming and the sphere–sphere case serves as a balanced reference. These formulas offer practical tools for estimating stresses in mill design and for validating computer simulations, supporting more efficient milling processes. • Analytical contact model for ball mills with ellipsoidal, spherical, and cylindrical jar geometries. • Explicit formulas derived for maximum force, pressure, deformation, and contact area during ball-jar impacts. • Opposite angular trends identified for oblate and prolate geometries: forces (pressure) peak at equator (poles) in oblate jars, and at poles (equator) in prolate jars.