Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties

The paper presents a fracture analysis for an electromagnetically dielectric crack in a functionally graded magnetoelectroelastic strip. It is considered that the material properties are varying exponentially along the width direction. Under the assumption of the in-plane magneto-electro-mechanical loadings, the dielectric crack is simulated by using the semi-permeable crack-face boundary conditions. The Fourier transform technique is applied to solve the boundary-value problem and four coupling singular integral equations are determined. A nonlinear system of algebraic equations is further derived and solved numerically to determine the electromagnetic field inside the crack. Then the field intensity factors of stress, electric displacement, and magnetic induction are given. Through the numerical computations, the effects of the material non-homogeneity and the permeability of crack interior on the electric displacement and the magnetic induction at the crack faces are studied. The variations of the intensity factors of stress, electric displacement, and magnetic induction versus the geometry of the crack, the strip width, and the material non-homogeneity are presented in graphics respectively.     

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.     

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     

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。     

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

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing an elliptic crack is considered. It is assumed that the body is under an external magnetic field perpendicular to the crack plane. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the elliptic crack are studied. Formulas are given for the stress intensity factors for the force and magnetic fields near the crack tip     

Crack problem for superconducting strip with finite thickness

The inclined crack problems are considered for a thin strip and a strip with finite thickness in a perpendicular magnetic field. The critical current density is assumed to be a constant. The crack orientation is varied and the effect of crack on the magnetic field distribution is neglected. Based on the analytical results and variational inequality, the field and current distributions are computed for both thin strip and strip with finite thickness cases, respectively. Then, the stress intensity factors at the crack tip are determined using the finite element method for magnetic field loads. The numerical results are presented for different inclined crack angles, magnetization processes and geometry parameters of the strip. The results show that the fracture behavior of the strip with finite thickness is more complicated than that of the thin strip. With the numerical results, we can predict the largest possibility of cracking as the strip is in an external field.     

Finite deformation theory for soft ferromagnetic elastic solids

A general theory of finite deformation of soft ferromagnetic elastic solids is formulated following the linear theory developed earlier by Pao and Yeh. The constitutive equations, field equations, and the boundary conditions of this theory are applied to analyse the buckling of a plate under the action of a uniform magnetic field. A nontrivial equilibrium configuration for the deformed plate is shown to exist, and the critical value of the externally applied magnetic induction at which the plate buckles is determined. It is demonstrated that the non-linear deformation affects the critical magnetic induction considerably.     

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO₃-CoFe₂O₄ composite with a penny-shaped vacuum crack are discussed in detail.     

承压热冲击下反应堆压力容器的塑性失效分析

承压热冲击（PTS）使反应堆压力容器（RPV）的完整性面临极大的挑战,尤其是在喷嘴周围的环带区。考虑到瞬态温度处于零韧性参考温度之上,引入非线性材料性能来模拟一个真实RPV的混合温度场和应力场。应用扩展有限元法对喷嘴区中的裂纹扩展过程进行模拟,并得到了承压热冲击下的临界裂纹尺寸。结果显示,PTS初期的热应力效应显著,后期热-机械耦合作用产生的峰值应力很可能会引起结构失效。应用直接和间接耦合法得到的数值结果吻合,且后者的计算效率较高。在塑性极限承载状况下,裂纹尖端在靠近内壁的位置容易产生扩展。离内壁较远的裂纹尖端由于受热冲击影响较小,发生裂纹扩展的可能性相对较低。随着基体墙厚度的减小,容许的裂纹尺寸急剧缩小,接近临界承载状态时稳态裂纹扩展的程度明显降低。     

Fracture mechanics for a mode III crack in a magnetoelectroelastic solid

In this paper, we developed a Stroh-type formalism for anti-plane deformation and then investigated the fracture mechanics for an elliptical cavity in a magnetoelectroelastic solid under remotely uniform in-plane electromagnetic and/or anti-plane mechanical loading, which allowed us to take the electromagnetic field inside the cavity into account. Reducing the cavity into a crack, we had explicit solutions in closed forms for a mode III crack, which included the extreme cases for an impermeable crack and a permeable crack. The results were illustrated with plots, showing that in the absence of mechanical loads, an applied electric or magnetic field, positive or negative, always tended to close the crack. On the other hand, in the presence of a mechanical load, a negative electric or magnetic field retarded crack growth, while a positive field could either enhance or retard crack propagation, depending on the strengths of the applied electric/magnetic fields and the level of the mechanical load as well. In other words, the effect of electric/magnetic fields on the fracture behavior is mechanical load-dependent.     

K III -Deformation modes in internal oblique cracks under plane-stress conditions

The exact shape of the deformed internal oblique crack in an infinite elastic plate under conditions of plane stress was studied. The Muskhelishvili potential function yielded the exact stress and displacement field around the crack. While in previous papers by the author the study was mainly concerned with the definition of the in-plane shape of the deformed crack and important properties were disclosed concerning especially the contribution of the shear loading of the crack, in this paper the out-of-plane component of displacements is determined and its influence on the exact shape of the deformed crack in space is presented. It is shown that, as soon as shear displacements appear in the cracked plate under plane stress, out-of-plane shear displacements are a compulsory consequence for plane-stress conditions of the plate. The elliptic form of the deformed internal crack was twisted out of the plane of the plate with its zero twisting displacement near the new crack tip of its deformed shape corresponding to the vertex of the ellipse. The points of maximum and minimum out-of-plane displacements were placed close to the vertices of the ellipse at polar angles, θ, depending only on the eccentricity of the ellipse and displaced always on both sides of the vertices. The compulsory coexistence of mode II and mode III deformations makes the internal crack in plane stress to present a complicated pattern of deformation at its deformed crack tips. All of these results are amply supported by experimental evidence with caustics, which always show either a simple mode I pattern or a complex mode II and III pattern as soon as shear interferes in the mode of deformation of the plate.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.     

Buckling and post-buckling of a ferromagnetic beam-plate induced by magnetoelastic interactions

Buckling and post-buckling behavior of a soft ferromagnetic beam-plate with unmovable simple supports under an applied magnetic field is analyzed by taking into account the non-linear effect of large deflections based on the von Karman model of plates. The study shows that buckling is proceeded by bending only if the applied magnetic field forms an incident angle with the normal of the plate. The characteristics of buckling and post-buckling are explored by numerical analysis. In particular, the numerical solution shows that, when the applied magnetic field is oblique, the plate snapps from a whole wave configuration into a half-wave configuration as buckling/snapping occurs, and that the magnetoelastic buckling strength increases with the increasing oblique angle.     

Loading effect on ACPD of a crack in ferromagnetic material

The loading effect on alternating current potential drop (ACPD) for a ferromagnetic material containing a two-dimensional surface crack was investigated under opening mode loading without shear (mode I). The change in potential drop due to load was obtained with and without a magnetic field around the specimen. To remove the magnetic field from the circumference of the specimen, a new measuring system using the characteristic of coaxial transmission line was made. When the magnetic field does not exist around the specimen, the change in potential drop with load is governed by the change in electromagnetic properties near the crack tip. The results obtained by using the new measuring system are the basis for an application of the ACPD technique to the experimental determination of the stress intensity factor, since they are independent of the arrangement of the measuring probe lines and the current supply lines. The relationship between the change in potential drop and the change in load is linearized by demagnetization. The change in potential drop per unit change in the stress intensity factor is independent of the crack length.     

Intensification of externally applied magnetic field around a crack in layered composite

Dynamic stress intensification around the crack in a layered composite subjected to an externally applied magnetic field is investigated. The effect of magnetic force is accounted for in the analysis as a body force. It tends to introduce damping into the system and reduces the intensification of the crack tip stress field. This is illustrated through the variations of the Mode I and Mode II stress intensity factors with time in addition to the influence of the geometric and material parameters of the composite structure. Numerical results are presented and discussed as the magnetic flux is varied with the physical parameters.     

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.     

The effect of magnetic field on a microgravity single droplet combustion

The effects of magnetic field on the microgravity combustion characteristics of a single methanol droplet in homogeneous flow are numerically investigated to develop an effective magnetic control method for microgravity droplet combustion and spray combustion systems. First, governing equations of microgravity single methanol droplet combustion under a homogeneous magnetic field based on an unsteady two-dimensional, spherically symmetric model including single-step chemistry are presented. Employing numerical modeling, several combustion behaviors are calculated taking into account the effect of the unsteady magnetic field profiles at the flame front. It is found that the flame front becomes deformed and is elongated in the direction of the magnetic field due to the inhomogeneous magnetic pressure distribution at the interface between the fuel vapor phase and the oxidizer phase. This nonuniformity of magnetic pressure is caused by the transient deformation of the magnetic field with refraction of magnetic flux at the flame front due to the difference of magnetic susceptibility between the diamagnetic fuel vapor phase and the paramagnetic oxidizer phase.     

Perturbed magnetic field of an infinite plate with a centered crack

Deforming a cracked magnetoelastic body in a magnetic field induces a perturbed magnetic field around the crack. The quantitative relationship between this perturbed field and the stress around the crack is crucial in developing a new generation of magnetism-based nondestructive testing technologies. In this paper, an analytical expression of the perturbed magnetic field induced by structural deforma- tion of an infinite ferromagnetic elastic plate containing a centered crack in a weak external magnetic field is obtained by using the linearized magnetoelastic theory and Fourier transform methods. The main finding is that the perturbed magnetic field intensity is proportional to the applied tensile stress, and is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangential component of the perturbed magnetic-field intensity near the crack exhibits an antisymmetric distribution along the crack that reverses its direction sharply across its two faces, while the normal component shows a symmetric distribution along the crack with singular points at the crack tips.     

On magneto-thermo-viscoelastic deformation and fracture

A thermodynamic formulation of magneto-thermo-viscoelastic constitutive and fracture theory is developed, accounting for non-linear, thermal and hysteresis effects. State equations and energy flux integral are obtained in consideration of the Helmholtz free energy including the contribution of the free magnetic field as a functional of the histories of deformation, temperature and magnetic induction in the reference configuration. The rate of energy flow towards the crack front per unit crack advance provides the crack driving force in the presence of magneto-thermo-mechanical coupling and hysteresis, which is evaluated through formation of energy-momentum tensor for steady-state crack propagation. A fracture criterion based on the generalized -integral thus formulated overcomes the difficulty encountered by existing treatments and helps to understand the fracture behavior of both conservative and dissipative materials subject to combined magnetic, thermal and mechanical loadings. Reduction of this formulation to finite magneto-thermo-viscoelasticity is provided with polynomial expansion of the Helmholtz free energy functional.