Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.     

Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium

The hyper-singular boundary integral equation method of crack analysis in three-dimensional transversely isotropic magnetoelectroelastic media is proposed. Based on the fundamental solutions or Green’s functions of three-dimensional transversely isotropic magnetoelectroelastic media and the corresponding Somigliana identity, the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces. The extended displacement discontinuities include the displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, and correspondingly the extended tractions on crack face represent the conventional tractions, the electric displacement and the magnetic induction boundary values. The near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach. A solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials. The influence of different electric and magnetic boundary conditions, i.e., electrically and magnetically impermeable and permeable conditions, electrically impermeable and magnetically permeable condition, and electrically permeable and magnetically impermeable condition, on the solutions is studied. The crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical-electric-magnetic loadings. An iteration approach is presented for the solution of the non-linear model. The exact solution is obtained for the case of uniformly applied loadings on the crack faces. Numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method.     

Transient response of a cracked magnetoelectroelastic strip under anti-plane impact

The transient anti-plane problem of a magnetoelectroelastic strip containing a crack vertical to the boundary is considered. Singular integral equations for the impermeable crack are obtained by using Fourier and Laplace transforms. Numerical results show the effects of the relative loading parameters κ_D and κ_B, and the crack configuration on the dynamic fracture behavior. The results obtained indicate that for the impermeable crack, the electric and magnetic impacts have significant influences on the dynamic stress intensity factor and the dynamic energy density factor.     

Dynamic fracture analysis of a penny-shaped crack in a magnetoelectroelastic layer

This paper analyzes the dynamic magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic layer subjected to prescribed stress or prescribed displacement at the layer surfaces. Two kinds of crack surface conditions, i.e., magnetoelectrically impermeable and permeable cracks, are adopted. The Laplace and Hankel transform techniques are employed to reduce the problem to Fredholm integral equations. Field intensity factors are obtained and discussed. Numerical results of the crack opening displacement (COD) intensity factors are presented and the effects of magnetoelectromechanical loadings, crack surface conditions and crack configuration on crack propagation and growth are examined. The results indicate that among others, the fracture behaviors of magnetoelectroelastic materials are affected by the sizes and directions of the prescribed magnetic and/or electric fields, and the effects are strongly dependent on the elastic boundary conditions.     

A periodic array of cracks in a transversely isotropic magnetoelectroelastic material

Magnetoelectroelastic materials are inherently brittle and prone to fracture. Therefore, it is important to evaluate the fracture behavior of these advanced materials. In this paper, a periodic array of cracks in a transversely isotropic magnetoelectroelastic material is investigated. Hankel transform is applied to solve elastic displacements, electric potential and magnetic potential. The problem is reduced into a system of integral equations. Both impermeable and permeable crack-face electromagnetic boundary conditions assumptions are investigated. Quantities of the stress, electric displacement and magnetic induction and their intensity factor are obtained. Effect of the crack spacing on these quantities is investigated in details.     

Fracture of magnetoelectroelastic composite materials using boundary element method (BEM)

The behavior of cracked linear magnetoelectroelastic solids is analysed by means of the dual Boundary Element Method (BEM) approach. Media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. An explicit 2-D Green’s function in terms of the extended Stroh formalism for magnetoelectroelastic full-plane under static loading is implemented. Hypersingular integrals arising in the traction boundary integral equations are computed through a regularization technique. Evaluation of fracture parameters directly from computed nodal values is discussed. The stress intensity factors (SIF), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated for different crack configurations in both finite and infinite solids subjected to in-plane combined magnetic–electric–mechanical loading conditions. The accuracy of the boundary element solution is confirmed by comparison with selected analytical solutions in the literature. The new results that can be of interest in the design and maintenance of novel magnetoelectroelastic devices are also discussed.     

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.     

Semi-permeable crack analysis in a 2D magnetoelectroelastic medium based on the EMPS model

For a crack in a magnetoelectroelastic plane under the electrically and magnetically semi-permeable boundary condition, we derive the non-linear analytical solution of the strip electric–magnetic polarization saturation (EMPS) model. Using the extended dislocation theory and integral equation method, we obtain the electric and magnetic yielding zones, as well as the field intensity factor and local J-integral. Adapting an iterative method, numerical examples were performed to analyze the effect of different boundary conditions and the electric–magnetic saturated properties on the electric displacement and magnetic induction in the crack cavity, electric and magnetic yielding zones, stress intensity factor and local J-integral.     

Dynamic fracture analysis of an annular interfacial crack between dissimilar magnetoelectroelastic layers

Transient response of an annular interfacial crack between dissimilar magnetoelectroelastic layers under impacts is investigated. On the crack surface, magnetoelectrically impermeable boundary condition is adopted. Using Laplace and Hankel transform techniques, the mixed boundary value problem is reduced to a system of singular integral equations. The integral equations are further reduced to a system of algebraic equations with the aid of Jacobi polynomials. The dynamic field intensity factor and dynamic energy release rate are determined. Numerical results reveal the effects of electric or magnetic loadings and material parameters of composite on crack propagation and growth.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

Crack tip field in piezoelectric/piezomagnetic media

This paper considers the magnetoelectroelastic problem of a crack in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects. Based on the extended Stroh formalism, the general two-dimensional solutions to the magnetoelectroelastic problem are obtained, involving five analytic functions of different variables. The magnetoelectroelastic field around the crack tip is given. It contains five modes of square root singularities. Expressions of the stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived and the field intensity factors are provided. The path-independent conservative integral is derived. The energy release rate is written in terms of those field intensity factors. The explicit algebraic results are given for a special case of an anti-plane crack in a magnetoelectroelastic medium.     

Dynamic analysis of a cracked magnetoelectroelastic medium under antiplane mechanical and inplane electric and magnetic impacts

The transient analysis of a magnetoelectroelastic medium containing a crack is made under antiplane mechanical and inplane electric and magnetic impacts. The crack is assumed to penetrate through the solid along the poling direction. By using the Fourier and Laplace transforms, the associated mixed boundary value problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. By means of a numerical inversion of the Laplace transform, dynamic field intensity factors are obtained in the time domain. Numerical results are presented graphically to show the effects of the material properties and applied electric and magnetic impacts on the dynamic intensity factors of COD and stress, and dynamic energy density factors. The results indicate that except for the intensity factors of electric displacement and magnetic induction, other field intensity factors exhibit apparent transient feature. Moreover, they depend strongly on mechanical input as well as electric and magnetic impacts.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

BIEM analysis of dynamically loaded anti-plane cracks in graded piezoelectric finite solids

Anti-plane cracks in finite functionally graded piezoelectric solids under time-harmonic loading are studied via a non-hypersingular traction based boundary integral equation method (BIEM). The formulation allows for a quadratic variation of the material properties in two directions. The boundary integral equation (BIE) system is treated by using the frequency dependent fundamental solution based on Radon transforms. Its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor (SIF) and the electrical displacement intensity factor (EDIF) are determined. Several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters.     

Magnetoelectroelastic analysis for an opening crack in a piezoelectromagnetic solid

Magnetoelectroelastic analysis of a cracked piezoelectromagnetic solid is made within the framework of the theory of linear magnetoelectroelasticity. The associated mixed boundary-value problem is solved by the Fourier integral transform. For general electromagnetic crack-face boundary conditions, a full magnetoelectroelastic field in the entire plane induced by a crack is obtained explicitly, and field intensity factors and energy release rate are given. The influences of applied electric and magnetic loadings on the energy release rate, the strain intensity factor, and the stress distribution are presented graphically.     

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.     

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.     

Theoretical model of crack branching in magnetoelectric thermoelastic materials

Thermomagnetoelectroelastic crack branching of magnetoelectro thermoelastic materials is theoretically investigated based on Stroh formalism and continuous distribution of dislocation approach. The crack face boundary condition is assumed to be fully thermally, electrically and magnetically impermeable. Explicit Green’s functions for the interaction of a crack and a thermomagnetoelectroelastic dislocation (i.e., a thermal dislocation, a mechanical dislocation, an electric dipole and a magnetic dipole located at a same point) are presented. The problem is reduced to two sets of coupled singular integral equations with the thermal dislocation and magnetoelectroelastic dislocation densities along the branched crack line as the unknown variables. As a result, the formulations for the stress, electric displacement and magnetic induction intensity factors and energy release rate at the branched crack tip are expressed in terms of the dislocation density functions and the branch angle. Numerical results are presented to study the effect of applied thermal flux, electric field and magnetic field on the crack propagation path by using the maximum energy release rate criterion.     

Strip electric-magnetic breakdown model in a magnetoelectroelastic medium

Extending the polarization saturation model [Gao et al., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491-510] and the dielectric breakdown (DB) model [Zhang et al., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311-327] in piezoelectric materials, the Strip Electric-Magnetic Breakdown (SEMB) model is proposed for electrically and magnetically impermeable crack in a magnetoelectroelastic medium to study the effect of the nonlinear character of electric field and magnetic field on fracture of magnetoelectroelastic materials. In the SEMB model, the electric field in the strip of the electric breakdown zone ahead of the crack tip is equal to the electric breakdown strength, while the magnetic filed in the strip of the magnetic breakdown zone is equal to the magnetic breakdown strength. By using the extended Stroh formalism and the extended dislocation modeling of a crack, the Griffith crack problem under the electrically and magnetically elastic-plastic condition in a magnetoelectroelastic medium is reduced to a set of dual integral equations. The sizes of the electric breakdown zone and the magnetic breakdown zone, the extended intensity factors and the local J-integral are obtained. The effect of the combined mechanical-electric-magnetic loadings on the local J-integral is studied.     

压电压磁复合材料中正六边形孔边裂纹的反平面断裂问题

基于线性磁电弹性理论,利用Schwarz-Christoffel(CS)变换技术和Stroth公式,首次系统研究了压电压磁复合材料中含带两个不对称裂纹的正六边形孔口问题在部分渗透磁电边界条件下的解析解.当忽略磁场时,磁电非渗透裂纹和磁电渗透裂纹两种极端情况下的解析解答可退化为文献已有研究结果.数值结果揭示了正六边形孔口尺寸、裂纹长度以及力电载荷和磁载荷对能量释放率的影响规律.研究结果表明：减小孔口边长和裂纹长度可以提高材料的可靠性;机械载荷总是促进裂纹扩展;在磁电非渗透和磁电部分渗透边界条件下,负电场和负磁场会延缓裂纹的扩展,而正电场可以增强或阻碍裂纹的扩展,这取决于所施加的电场和磁场的强度以及机械载荷的水平;在磁电渗透边界条件下,电场和磁场对裂纹的扩展没有影响.