General Formulation for Magnetic Forces in Linear Materials and Permanent Magnets

A previous work described how a magneto-mechanical tensor can be obtained from the application of the virtual work principle to the energy density or co-energy density in any given media. This paper describes how the volume and surface force densities can be obtained from this tensor. Taking these force densities into account, we can calculate the total force over any given magnetic medium that is surrounded by any other magnetic medium. This represents a generalization of the classic Maxwell theorem. With this methodology, it is possible to calculate the surface forces in bodies in contact. We demonstrate this by showing the equivalence of the results obtained by this procedure and the equivalent source method and the Kelvin expression method. These methods could previously be applied only to bodies surrounded by air. We also show that this procedure is equivalent to other methods used to calculate the forces between linear magnetic materials (Korteweg–Helmholtz's tensor, ponderomotive force density). We have compared our method with results from continuum design sensitivity analysis, a method that allows for the calculation of global and local forces in magnetic media in contact .