Analysis of cracked magnetoelectroelastic composites under time-harmonic loading

This paper presents a numerical model for the analysis of cracked magnetoelectroelastic materials subjected to in-plane mechanical, electric and magnetic dynamic time-harmonic loading. A traction boundary integral equation formulation is applied to solve the problem in combination with recently obtained time-harmonic Green’s functions (Rojas-Diaz et al., 2008). The hypersingular boundary integral equations appearing in the formulation are first regularized via a simple change of variables that permits to isolate the singularities. Relevant fracture parameters, namely stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor are directly evaluated as functions of the computed nodal opening displacements and the electric and magnetic potentials jumps across the crack faces. The method is checked by comparing numerical results against existing solutions for piezoelectric solids. Finally, numerical results for scattering of plane waves in a magnetoelectroelastic material by different crack configurations are presented for the first time. The obtained results are analyzed to evaluate the dependence of the fracture parameters on the coupled magnetoelectromechanical load, the crack geometry and the characteristics of the incident wave motion.     

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.     

Stress state of a transversely isotropic magnetoelectroelastic body with a plane crack under antisymmetric loads

The static equilibrium of a transversely isotropic magnetoelectroelastic body with a plane crack of arbitrary shape in the isotropy plane under antisymmetric mechanical loading is studied. The relationships between the stress intensity factors (SIFs) for an infinite magnetoelectroelastic body and the SIFs for a purely elastic body with the same crack and under the same antisymmetric loading are established. This enables the SIFs for a magnetoelectroelastic body to be found directly from the analogous problem of elasticity. As an example of using this result, the SIFs for penny-shaped, elliptic, and parabolic cracks in a magnetoelectroelastic body under antisymmetric mechanical loading are found Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 37–51, October 2008.     

电磁弹性固体三维问题的虚边界元-等额配点法

依据弹性力学虚边界元法的基本思想和电磁弹性固体的基本解,提出了电磁弹性固体三维问题的虚边界元-等额配点法.该方法继承传统边界元法优点的同时,有效地避免了传统边界元法的边界积分奇异性的问题.算例表明该方法有很高的精度,是求解电磁弹性固体三维问题的一个有效的数值方法.     

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.     

Interface crack with a contact zone in an isotropic bimaterial under thermomechanical loading

A plane problem for a thermally insulated interface crack with a contact zone in an isotropic bimaterial under tension–shear mechanical loading and a temperature flux is considered. The expressions for the stresses and the electrical flux as well as for the derivatives of the displacement and the temperature jumps at the material interfaces via sectionally holomorphic mechanical and thermal potential functions are given. After the solution of the thermal problem the inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved exactly. The stresses at the interface and the stress intensity factors at the singular points are presented in a clear analytical form. Special attention is devoted to the case of a small contact zone when the stress intensity factors can be presented in form similar to the associated presentation for an “open” crack model. A transcendental equation and an asymptotic analytic formula for the determination of the real contact zone length are derived. It is shown that for a certain bimaterial this length as well as the correspondent stress intensity factor are defined by a single parameter which depends on the normal-shear loading and the heat flux.     

磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.      

Anti-plane analysis of multiple cracks originating from a circular hole in a magnetoelectroelastic solid

The anti-plane problem of multiple cracks originating from a circular hole in a magnetoelectroelastic solid is investigated under remotely uniform anti-plane mechanical loading and in-plane electromagnetic loadings. The boundary value problem is reduced to a Cauchy integral equation by a new mapping function and the complex variable method, which is further solved exactly. The analytic expressions of the complex potentials, the field intensity factors and the energy release rate are derived in closed-form under the assumption that the surfaces of the cracks and hole are both electrically and magnetically impermeable. The effects of crack configurations and combined loadings on the energy release rate are shown graphically. Several useful results which may have potential applications to the design and fracture analysis of magnetoelectroelastic structures are given.     

THREE-DIMENSIONAL ANALYSIS OF A MAGNETOELECTROELASTIC HALF-SPACE UNDER AXISYMMETRIC TEMPERATURE LOADING 1)

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.     

Transient response of an interfacial crack between dissimilar magnetoelectroelastic layers under magnetoelectromechanical impact loadings: Mode-I problem

The dynamic response of an interfacial crack between two dissimilar magnetoelectroelastic layers is investigated under magnetic, electrical and mechanical impact loadings. Four kinds of ideal crack-face assumptions, i.e., magnetoelectrically impermeable (Case 1), magnetically impermeable and electrically permeable (Case 2), magnetically permeable and electrically impermeable (Case 3) and magnetoelectrically permeable (Case 4), are adopted separately. The dynamic field intensity factors and energy release rates are derived. The effects of loading combinations and crack configurations especially for the former on the dynamic response are examined according to energy release rate criterion. The numerical results show that, among others, a negative magnetic (or electrical) loading is generally prone to inhibit the crack extension rather than a positive one for a magnetically (or electrically) impermeable interfacial crack. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures.     

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.     

Finite boundary effects in problem of a crack perpendicular to and terminating at a bimaterial interface

In this paper, the problem of a crack perpendicular to and terminating at an interface in bimaterial structure with finite boundaries is investigated. The dislocation simulation method and boundary collocation approach are used to derive and solve the basic equations. Two kinds of loading form are considered when the crack lies in a softer or a stiffer material, one is an ideal loading and the other one fits to the practical experiment loading. Complete solutions of the stress field including the T stress are obtained as well as the stress intensity factors. Influences of T stress on the stress field ahead of the crack tip are studied. Finite boundary effects on the stress intensity factors are emphasized. Comparisons with the problem presented by Chen et al. (Int. J. Solids and Structure, 2003, 40, 2731–2755) are discussed also.The project supported by the National Natural Science Foundation of China (10202023 and 10272103), and the Key Project of CAS (KJCX2-SW-L2).     

A PENNY-SHAPED CRACK IN A MAGNETOELECTROELASTIC LAYER UNDER RADIAL SHEAR IMPACT LOADING

This paper analyzes the dynamic magnetoelectroelastic behavior induced by a penny- shaped crack in a magnetoelectroelastic layer.The crack surfaces are subjected to only radial shear impact loading.The Laplace and Hankel transform techniques are employed to reduce the prob- lem to solving a Fredholm integral equation.The dynamic stress intensity factor is obtained and numerically calculated for different layer heights.And the corresponding static solution is given by simple analysis.It is seen that the dynamic stress intensity factor for cracks in a magnetoelec- troelastic layer has the same expression as that in a purely elastic material.And the influences of layer height on both the dynamic and static stress intensity factors are insignificant as h/a>2.     

A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.     

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.     

A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading

The solution for an elliptical cavity in an infinite two-dimensional magnetoelectroelastic medium subject to remotely uniformly applied combined mechanical–electric–magnetic loadings is obtained by using the Stroh formalism and the exact boundary conditions along the surface of the cavity. By letting the minor-axis of the cavity to zero the solution for a crack is deduced. A self-consistent method is proposed to calculate the real crack opening under the combined mechanical–electric–magnetic loadings. The method requires that the crack opening is the minor-axis of the elliptical opening profile. Beside the real crack solution, four different extreme models, i.e., the impermeable crack, permeable crack, electrically impermeable and magnetically permeable crack and electrically permeable and magnetically impermeable crack, are discussed. An expression of the strain energy density factor is derived. Numerical results of the strain energy density at the crack tip are given for a BaTiO₃–CoFe₂O₄ composite with the piezoelectric BaTiO₃ material being the inclusion and the magnetostrictive CoFe₂O₄ material being the matrix. The effects of the proportion of the two phases, permeability of the crack to electric and magnetic fields, the electric and magnetic loadings on the strain energy density factor are discussed.     

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.     

Micromechanics predictions of the effective moduli of magnetoelectroelastic composite materials

In this paper, the self-consistent, generalized Mori–Tanaka and dilute micromechanics theories are extended to study the coupled magnetoelectroelastic composite materials. The heterogeneous inclusion problem of magnetoelectroelastic behavior is formulated in terms of five interaction tensors related to the Green's functions for an infinite three-dimensional transversely isotropic magnetoelectroelastic solid. These tensors are then used to predict the effective moduli of the magnetoelectroelastic solid based on the self-consistent, Mori–Tanaka and the dilute approaches. Numerical results are obtained for various types of inclusions. These results are employed to study the effects of the inclusion properties, such as moduli, volume fractions, shapes, etc., on the effective moduli of magnetoelectroelastic composites, in particular, the related magnetic properties. The results obtained using the self-consistent model, the generalized Mori–Tanaka's model and the dilute approach are compared with the existing experimental and theoretical results.     

Spontaneous Fracture of Solids as Affected by Surfactants

Fracture of solids under the action of surfactants is studied. The limiting case of manifestation of chemisorption effects without significant effect of mechanical stresses is considered. The growth of plane cracks in solids with a regular structure upon contact with surfactants is described for the case where the model of the process is molecular wedge–assisted crack wedging. The critical stress–intensity factors are estimated in terms of the critical crack opening in wedging by a semi–infinite wedge of constant thickness. An anomalously low resistance to breaking of a solid into parts is observed under certain conditions (spontaneous fracture).     

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.