Magnetoelastic fracture of soft ferromagnetic materials
Introduction
Wide spread engineering applications of soft ferromagnetic materials (SFM) have attracted the attention of many researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] concerning the magnetoelastic deformation and fracture of soft ferromagnetic structures subjected to external magnetic field. In order to explore the magnetoelastic fracture of the soft ferromagnetic steel under magnetic field, which is usually employed in the devices of nuclear reactor, the work in [15] conducted a test with compact tension (CT) specimens of the ferromagnetic alloy Incoloy 908 in order to find the impact of external magnetic field on fracture toughness. However, the results were not repeated. By employing the integral transform technique and the linearized model developed in [8], the stress field near a crack-tip in an infinite soft ferromagnetic medium subjected to an external magnetic field perpendicular to the crack line was derived [16]. This method was further extended to obtain the solutions of many theoretical models of the ferromagnetic body with a crack in the presence of magnetic field [17], [18], [19], [20], [21]. Derived in [22] was the magnetic potential, the magnetoelastic and Maxwell stresses in the anisotropic soft ferromagnetic half-space. Other models of magnetoelasticity in a half-space were reported in [23], [24], [25] for the effect of electromagnetic force on the crack-tip stress field. It was pointed out that the of electric current around the crack-tip is also singular. Based on the rotationally-invariant (finite-strain) quasi-magnetostatic theory, the expressions for energy-release rates were established and the path-independent integrals were constructed for the elastic soft ferromagnets [26] and hard ferromagnets [27]. Developed in [28] is a theory, using a linearized model of [8]. The complex variable method was used to obtain a solution of the stress field near a collinear crack in an infinite soft ferromagnetic plane. Investigated in [29] is the effect of magnetic field and boundary conditions on the magnetoelastic stress intensity factor in a half-plane under an external magnetic field.
A soft ferromagnetic body in a magnetic field will exhibit magnetostrictive behavior and, in the mean time, it will be subjected to a magnetic force. In the above-mentioned literature, the magnetostrictive behavior was generally not considered when the deformation and fracture of soft ferromagnetic materials were investigated. Only the deformation originated from the magnetic force was considered to be the effect of the magnetic field on the material. However, some magnetic materials do exhibit significant magnetostrictive effects, such as the compound series of rare earth elements and iron, which have wide applications due to their enormous magnetostriction. The effect of magnetostriction is vital to the deformation of this kind of materials [30]. Generally, the stress field in the neighborhood of the tip of a crack-like flaw is determined by both the magnetic force and the magnetostrictive effect of soft ferromagnetic medium. In this paper, we present some studies on the magnetoelastic fracture of the magnetostrictive materials by means of experimental techniques and theoretical models. To attain experimental results of magnetoelastic fracture on soft ferromagnetic materials, the manganese–zinc ferrite ceramic were tested for the variation of fracture toughness under magnetic field with the three-point bending method and the Vickers’ indentation techniques [31]. Three analytical models were presented addressing the magnetoelastic fracture of the magnetostrictive materials, i.e. the anti-plane shear model, the in-plane fracture model with an elliptical crack, and the small-scale magnetic-yielding model. These theoretical models can be used to qualitatively interpreting the experimental results of fracture of the manganese–zinc ferrite ceramics under magnetic field [31] and the fracture test on the ferromagnetic alloy Incoloy 908 with compact tension specimens obtained in [15].
Section snippets
Fracture experiments of manganese–zinc ferrite ceramic
The manganese–zinc ferrite ceramics was selected to explore the effects of magnetic field on fracture toughness of soft ferromagnetic materials with the single-edge-notch-bend (SENB) specimens [31]. For the sake of generality, three groups of ceramics with different permeability were specially selected in the series of manganese–zinc ferrite. Furthermore, the Vickers’ indentation was also conducted on the polished ceramic samples with an attempt to find the variation of fracture toughness both
The anti-plane shear
The anti-plane shear problem is the simplest mode in fracture mechanics, which is usually selected for theoretical analysis when complex issues involving various coupled fields are addressed. In what follows, the anti-plane is solved for the magnetoelasticity problem where an in-plane magnetic field, H∞, and an out-of-plane shear stress, , are applied at infinity of an infinite magnetostrictive body with a finite center-through crack of length of 2a, as shown in Fig. 5. The applied magnetic
Strain energy density factor (SEDF) for the anti-plane shear model
The direct reason of fracture is the mechanical stress. The failure criterion of strain energy density factor, S, developed by Sih [43] is adoptedwhere Sm is the strain energy density factor with magnetoelasticity. is the strain energy density. The strain energy density factor can be obtained by substituting stress field equations into Eq. (35)where θ0, G2, A(θ) are referred to Eqs. (15), (11). In terms of the failure criterion of SEDF, the crack
Conclusions
Presented are studies concerned with the magnetoelastic fracture of the magnetostrictive materials, including the experimental results and several theoretical models. The manganese–zinc ferrite ceramics were tested for the variations of fracture toughness under magnetic field with the three-points bending method and the Vickers’ indentation techniques. An anti-plane shear model was considered for the magnetoelasticity of an infinite body of magnetostrictive material with a finite center-through
Acknowledgment
Support from the National Science Foundation of China under grants #10025209, #10132010, #10102007, #90208002 and from the Research Grants Council of the Hong Kong Special Administrative Region, China (RGC, Project No. HKU 7063/01E) is acknowledged. The authors also are grateful for the support by the Key grant Project of Chinese Ministry of Education (0306).
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