Magnetoelastic fracture of soft ferromagnetic materials

Effects of magnetic field on fracture toughness of soft ferromagnetic materials were studied using experimental techniques and theoretical models. The manganese–zinc ferrite with a single-edge-notch-beam (SENB) were chosen to be the specimen and the Vickers’ indentation specimen subjected to a magnetic field were chosen to be the specimens. Results indicate that there is no significant variations of the measured fracture toughness of the manganese–zinc ferrite ceramic in the presence of the magnetic field. The theoretical model involves an anti-plane shear crack with finite length in an infinite magnetostrictive body where an in-plane magnetic field prevails at infinity. Magnetoelasticity is used. The crack-tip elastic field is different from that of the classical mode III fracture problem. Furthermore, the magnetoelastic fracture of the soft ferromagnetic material was studied by solving the stress field for a soft ferromagnetic plane with a center-through elliptical crack. The stress field at the tip of a slender elliptical crack is obtained for which only external magnetic field normal to the major axis of the ellipse is applied at infinity. The results indicate that the near field stresses are governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter for finding whether magnetostriction or magnetic-force-induced deformation is dominant near the front an elliptically-shaped crack. The influence of the magnetic field on the apparent toughness of a soft ferromagnetic material with a crack-like flaw can be regarded approximately in two ways: one possesses a large induction magnetostrictive modulus and the other has a small modulus. Finally, a small-scale magnetic-yielding model was developed on the basis of linear magnetization to interpret the experimental results related to the fracture of the manganese–zinc ferrite ceramics under magnetic field. Studied also is the fracture test of the soft ferromagnetic steel with compact tension specimens published in the existing literature.     

Magnetoelastic interactions in a soft ferromagnetic body with a nonlinear law of magnetization: Some applications

Linearized equations and boundary conditions of a magnetoelastic ferromagnetic body are obtained with the nonlinear law of magnetization. Magnetoelastic interactions in a multi-domain ferromagnetic materials are considered for magneto soft materials, i.e. the case when the magnetic field intensity vector and magnetization vector are parallel. As a special case, the following two problems are considered: (1) the magnetoelastic stability of a ferromagnetic plate-strip in a homogeneous transverse magnetic field; (2) the stress–strain state of a ferromagnetic plane with a moving crack in a transverse magnetic field. It is shown that the modeling of magnetoelastic equations with a nonlinear law of magnetization provides qualitative and quantitative predictions on physical quantities including critical loads and stresses. In particular, it is shown that the critical magnetic field in plate stability problems found with the nonlinear law of magnetization is in better agreement with the experimental finding than the one found with a linear law. Furthermore, it is also shown that the stress concentration factor around a crack predicted with the nonlinear law of magnetization is more accurate than the one obtained with a linear counterpart. Numerical results are presented for above mentioned two problems and for various forms of nonlinear laws of magnetization.     

Nonlinear magnetoelastic coupling effects in a soft ferromagnetic material with a crack

In this paper, the magnetoelastic coupling effect in an infinite soft ferromagnetic material with a crack is analyzed. The nonlinear effect of magnetic field upon stress and the effect of the deformed crack configuration are taken into consideration. The coupling field is determined in the deformed configuration by regarding the deformed crack as an elliptical cylinder with its geometric coefficients, which are determined from a set of algebraic equations deduced from the displacements. The magnetic and stress fields near the crack tip are discussed for the case where both of the magnetic loading and the mechanical tension are present.     

Stress State of a Transversely Isotropic Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The magnetoelastic stress-strain problem for a transversely isotropic ferromagnetic body with an elliptical crack in the isotropy plane is solved explicitly. The body is in an external magnetic field perpendicular to the isotropy plane. The magnetic field induces elastic strains and an internal magnetic field in the body. The main characteristics of stress-strain state and induced magnetic field are determined and their features in the neighborhood of the crack are analyzed. Formulas for the stress intensity factors of the mechanical and magnetic fields near the crack tip are presented__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 48–59, January 2005.     

软铁磁材料平面裂纹问题的耦合场

由磁弹性问题的线性化理论导出磁场下平面软铁磁体问题的控制方程和复势解。利用复势解和奇异积分方程方法,对面内磁场和远场载荷作用下的含裂纹无限大软铁磁平面问题进行了求解,得到耦合场的解。并对不同磁力模型的结果和磁场与机械载荷共同作用下的裂尖应力强度因子进行了讨论。     

Magneto-elastic stress in a thin infinite plate with an elliptical hole under uniform magnetic field

Two-dimensional magnetic field and magneto-elastic stress solutions are presented for a magnetic material of a thin infinite plate with an elliptical hole under uniform magnetic field. The linear constitutive equation is used for the magnetic field and the stress analyses. The magneto-elastic stress is analyzed using Maxwell stress since only Maxwell stress is caused as a body force according to the electro magneto theory. Except the approximation of the plane stress state in which the plate is thin, no further assumption is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress for soft ferromagnetic material is analyzed and then those for paramagnetic and diamagnetic materials are analyzed. It is stated that the stress components are the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields are different each other in the plates. If the analysis of magnetic field of paramagnetic materials is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material. Shear deflection as well as stress in the direction of the plate thickness arises and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived.     

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

Two dimensional solutions of the 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 subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. 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, 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 those plane 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. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

Interfacial crack problem in layered soft ferromagnetic materials in uniform magnetic field

The magnetoelastic plane strain problem of an interfacial Griffith crack between two dissimilar soft ferromagnetic elastic materials subjected to a uniform magnetostatic field is considered within the framework of linear magnetoelasticity. By making use of the Fourier integral transform technique, the mixed boundary problem is then reduced to a pair of singular integral equations of the second kind. Solutions of the singular integral equations are obtained numerically by means of a Jacobi polynomial expansion method. Effects of the magnetic field, the combinations of the magnetic properties of materials and the geometric parameters on the magnetoelastic stress intensity factors in the vicinity of crack tip are shown graphically.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

Equivalent thickness conception for corner cracks

Thickness dependence of the one-parameter-based fracture toughness has been well recognized and widely studied. However, it is still a challenge to predict the fracture of structures with curved cracks from the fracture toughness data obtained from the standard through-the-thickness cracked specimens. The complicated three-dimensional (3D) stress fields near the crack front play a vital role in the fracture strength of materials. Based on a systematical numerical study of the 3D stress fields near the crack tip of quarter elliptic corner cracks and comparison with that of ideal through-the-thickness cracks, an equivalent thickness conception for curved cracks is proposed from the viewpoint of out-of-plane constraint, and a semi-analytical solution for the equivalent thickness of corner cracks is obtained. With the evaluated equivalent thickness, the fracture toughness of corner cracked specimens is predicted efficiently by the plane-strain toughness value of the material obtained from the standard through-the-thickness specimen.     

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

Two-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.     

ANTI-PLANE ANALYSIS FOR ELLIPTICAL INCLUSION IN MAGNETOELECTROELASTIC MATERIALS

This paper considers the multi-field coupling in magneroelectroelastic composite materials consisting of the inclusion and the matrix are magnetoelectroelastic materials. The mechanical,electric and magnetic fields around an elliptical cylinder inclusion are formulated by complex potentials. Inside the inclusion,the strain,electric and magnetic fields are found to be uniform and vary with the shape of the ellipse. When the inclusion is reduced to a crack,along the interface,the strain,electric field strength and magnetic field strength equal the corresponding remote ones,which can be used as the boundary condition. Special cases,such as a rigid and permeable inclusion,a soft and impermeable inclusion,a line inclusion and a crack problem are discussed in detail.     

ASYMPTOTIC SOLUTIONS OF MODE I STEADY GROWTH CRACK IN MATERIALS UNDER CREEP CONDITIONS

An asymptotic analysis is made on problems with a steady-state crack growth coupled with a creep law model under tensile loads. Asymptotic equations of crack tip fields in creep materials are derived and solved numerically under small scale conditions. Stress and strain functions are adopted under a polar coordinate system. The governing equations of asymptotic fields are obtained by inserting the stress field and strain field into the material law. The crack growth rate rather than fracture criterion plays an important role in the crack tip fields of materials with creep behavior.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

压电材料裂纹顶端条状电饱和区模型的力学分析

在线性压电本构方程框架下,对裂纹顶端条状电饱和区模型进行了严格的数学分析．完整地考虑了各向异性力电耦合效应．建立了电饱和区尺寸与外加电场的依赖关系．证实了当裂纹垂直极化轴时,压电材料的断裂应力随着外加正电场的增加而减小,随着外加负电场的增加而增加．当裂纹平行于极化轴时,与极化轴平行的外加电场对断裂应力无影响     

层状磁电复合材料界面共线裂纹平面问题分析

本文研究了面内电磁势载荷作用下双层压电压磁复合材料中共线界面裂纹问题．考虑了压电材料的导磁性质和压磁材料的介电性质,引入了界面电位移和磁感强度的连续性条件．利用Fourier 变换得到一组第二类Cauchy 型奇异积分方程．进一步导出了相应问题的应力强度因子、电位移强度因子和磁感强度强度因子的表达式,给出了应力强度因子的数值结果．结果表明电磁载荷会导致界面裂纹尖端I、II 混合型应力奇异性,同时还伴随着电位移和磁感强度的奇异性．比较了双裂纹左右端的应力强度因子,发现在面内极化方向上施加面内磁势载荷时共线裂纹内侧尖端区域的两个法向应力场发生互相干涉增强．     

The Stress State of a Transversely Isotropic Ferromagnetic with a Parabolic Crack in a Homogeneous Magnetic Field

The magnetoelastic problem for a transversely isotropic ferromagnetic body with a parabolic crack in the plane of isotropy is solved explicitly. The body is in an external magnetic field, which is perpendicular to the plane of isotropy. The field induces elastic strains and a magnetic field in the body. The characteristics of the stress–strain distribution and induced magnetic field are determined; and their singularities in the neighborhood of the crack are analyzed. Formulas for the stress intensity factors of the mechanical and magnetic fields near the crack tip are presented     

A real-space phase field model for the domain evolution of ferromagnetic materials

A real-space phase field model based on the time-dependent Ginzburg–Landau (TDGL) equation is developed to predict the domain evolution of ferromagnetic materials. The phase field model stems from a thermodynamic theory of ferromagnetic materials which employs the strain and magnetization as independent variables. The phase field equations are shown to reduce to the common micromagnetic model when the magnetostriction is absent and the magnitude of magnetization is constant. The strain and magnetization in the equilibrium state are obtained simultaneously by solving the phase field equations via a nonlinear finite element method. The finite-element based phase field model is applicable for the domain evolution of ferromagnetic materials with arbitrary geometries and boundary conditions. The evolution of magnetization domains in ferromagnetic thin film subjected to external stresses and magnetic fields are simulated and the magnetoelastic coupling behavior is investigated. Phase field simulations show that the magnetization vectors form a single magnetic vortex in ferromagnetic disks and rings. The configuration and size of the simulated magnetization vortex are in agreement with the experimental observation, suggesting that the phase field model is a powerful tool for the domain evolution of ferromagnetic materials.