Two-dimensional problems in a soft ferromagnetic solid with an elliptic hole or a crack

This paper deals with the problem of an elliptic hole or a crack in a soft ferromagnetic solid under the magnetic field at infinity. First, based on the simplified versions of the linear theory of Pao and Yeh (1973), the general solution of an elliptical hole is obtained according to exact boundary conditions at the rim of the hole. Then, when the hole degenerates into a crack, explicit solutions are given for potential functions and intensity factors of total stresses. In the above analysis, three kinds of magnetic boundary conditions, that is, magnetically permeable, impermeable and conducting boundary condition, are considered on the surface of the hole or the crack, respectively. It is found that in general, the total stresses always have the classical singularity of the r-1/2r-1/2-type at the crack tips for considered three crack models, and that the applied magnetic field may either enhance or retard crack growth depending on the magnetic boundary conditions adopted on the crack faces, and the Maxwell stresses on the crack faces and at infinity. Since the present solutions for a crack are given in explicit form, they can also serve as a benchmark to test the validity of various analysis approaches or assumptions to more complicated crack problems in a soft ferromagnetic solid.