The development of a new generation of more efficient electric motors leads to designs with higher stresses, currents and electromagnetic fields. To improve on the prevailing existing methodology for the concurrent calculation of electromagnetic and mechanical fields in electric motors, the authors recently presented in Hanappier et al. (2021a) a multiphysics formulation of the problem using the direct (current configuration) approach of continuum mechanics together with analytical solutions of idealized motor problems. However, due to the complex geometry of a typical electric motor and the nonlinearity of the coupled (magneto-mechanical) constitutive laws, numerical solutions of the governing equations are required. To this end, a Lagrangian (reference configuration) variational principle is proposed for the eddy current approximation that properly retrieves the Maxwell stresses and is consistent with its direct approach counterpart. Based on this variational principle, a numerical (FEM) approach is proposed. It is next applied to an idealized (cylindrical) stator, where an analytical solution can be found for the linear magnetization regime, thus providing firstly an independent code verification and then an assessment of the influence of the stator's nonlinear magnetic response. The approach is subsequently used to tackle a realistic geometry stator with two pole pairs under a three-phase current for two different cases: loosely or tightly packed conducting wires to calculate the corresponding magnetic, stress and strain fields.