Extended Formulas to Compute Resultant and Contact Electromagnetic Force and Torque From Maxwell Stress Tensors

Force density calculus in permanent magnets and other nonlinear magnetic media is still a challenge despite two hundred years of electromagnetic theory. While generally accepted formulas to compute the total electromagnetic force on a domain exist in the literature, calculating the local distribution of forces that are needed, for instance, to couple electromagnetic and mechanical finite-element method simulations is a more open subject. In particular, it is widely believed that the contact force between magnetic materials touching each other cannot be properly estimated and the insertion of thin virtual air gaps is required. In this paper, first, the formulas existing in the bibliography to compute the resultant electromagnetic force and torque on a bounded domain are revisited and extended. The new formulas can be applied to cases where the boundary of the domain is a discontinuity surface for the magnetic field. In particular, they allow computing the force and torque on a magnet totally or partially embedded in a ferromagnetic material without using any artificial air layer between them. Then, these formulas are checked by using numerical simulation of suitable experiments.