Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Abstract

a b s t r a c tTwo-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H x , H y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H x , H y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

“…In the previous paper (Hasebe, 2010a), the electric current terms exist on the right hand side of (3a, b) which is different from the present paper. In material 1, (3) and (4) must be also satisfied.…”

“…5 in the present paper is the same form as that of the previous paper (electric current) (Hasebe, 2010a) except for the sign of H x , H y , H h because the direction of the magnetic field is different by 180°each other. The magnetic field H ð2Þ z is constant in the entire plate (see (18)) and is different from that of the electric current (Hasebe, 2010a).…”

“…The stress analysis is carried out in section 7. This shear deflection does not arise in a plate under uniform electric current in the previous paper (Hasebe, 2010a). It is necessary to state that the out of plate bending due to the bending moment is not caused by Maxwell stress shown in Fig.…”

“…The discontinuous magnitude of Maxwell stress between materials 1 and 2 is expressed as follows (Hasebe, 2010a):…”

“…In the previous paper (Hasebe, 2010a), the plate applied magnetic field induced by electric current was analyzed. Uniform magnetic field is one of the typical magnetic field as well as magnetic field caused by electric current.…”