Analytic and numeric solution of a magneto-mechanical inclusion problem

In this paper, exact solutions for the primary field variables and their first derivatives are derived for the coupled magneto-mechanical problem of a circular inclusion embedded in an infinite surrounding. Both material domains possess linear and isotropic material properties. The assumed planarity of all fields enables and recommends the description with analytic functions depending on the complex variable z. The well-known technique of a complex stress potential satisfying the bipotential equation is used and adapted to coupled magneto-mechanical problems. The provided analytic expressions are of special interest for the validation of numeric solutions of coupled magneto-mechanical boundary value problems. In this contribution, they are applied to analyze the convergence behavior of an extended finite element formulation for coupled magneto-mechanical problems. Based on the analytic results, proper boundary conditions are prescribed to the numeric model. The convergence in terms of polynomial degree and mesh refinement of the implemented element formulation is proved by computing the L2 and the energy norm which requires the knowledge of the exact solution. The generality of the proposed formulas allows for the deduction of some other kind of problems, e.g., the pure mechanical displacement and stress field of a bimaterial setting or the stress concentration around a circular hole.