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.     

Semi-permeable Yoffe-type interfacial crack analysis in MEE composites based on the strip electro-magnetic polarization saturation model

Fracture analysis of a semi-permeable Yoffe-type interfacial crack propagating subsonically in magneto-electro-elastic(MEE) composites is presented based on the strip electromagnetic polarization saturation(SEMPS) model. The electro-magnetic fields inside the crack are considered under the semi-permeable boundary condition. Nonlinear effects near the interfacial crack tip are represented by different electro-magnetic saturation zones. Utilizing the extended Stroh's method, we derive the moving dislocation densities as well as intensity factor and energy release rate for Yoffe-type MEE interfacial crack. Numerical results through an iterative approach are presented to show the characteristics of fracturedominant parameters with respect to propagation velocity and boundary condition category. The fracture-dominant parameters under the semi-permeable boundary condition are lower than those under the impermeable one, which implies that the electro-magnetic fields in the crack gap can retard the propagation of MEE interfacial crack.     

Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media

The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25–R29; Gao, H., Zhang, T.Y., Tong, P., 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, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311–327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for center cracks in piezoelectric strips. The effects of finite domain size, saturation property and different electric boundary conditions, as well as different models on the electric yielding zone and the local J-integral, are studied.     

Modeling of pre-fracture zones for limited permeable crack in piezoelectric materials

A plane-strain problem for a limited permeable crack in an adhesive thin interlayer between two semi-infinite piezoelectric spaces is considered. The tensile mechanical stress and the electric displacement are applied at infinity. The interlayer is assumed to be softer than the connected materials; therefore, the zones of mechanical yielding and electric saturations can arise at the crack tips on the continuations of the crack. These zones are considered in this work. It was assumed that the length of electric saturation zones is larger than the length of mechanical yield zones. The zones of mechanical yielding are modeled by the crack continuations with normal compressive stresses applied at its faces. The electric saturation zones are modeled by segments at the crack continuations with prescribed saturated electric displacements. These electric displacements can linearly vary along the mechanical yielding zones. The problem is reduced to the Hilbert–Riemann problem of linear relationship, which is solved exactly. The equation for the determination of the yielding zones length, the expressions for the crack-opening displacement jump, electric potential jump, and J-integral is obtained in an analytical form. In case of finite size body, the finite elements method is used and the variation in the fracture mechanical parameters with respect to this size is demonstrated.     

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     

Modified strip saturated models for two equal collinear cracks with coalesced zones in piezoelectric media

Two equal collinear cracks with coalesced interior electric saturation zones are analytically studied for two-dimensional(2D) arbitrary polarized semipermeable piezoelectric media based on a modified strip saturated model. The strip saturated model is modified here by varying the strip saturated constant electric displacement condition to the polynomially varying electric displacement conditions. Based on the linear, quadratic,and cubic electric displacement conditions on the inner and outer saturated zones, different modified strip saturated models are proposed and studied for two equal collinear cracks. With the Stroh formalism and the complex variable technique, these fracture problems are reduced into different types of non-homogeneous Riemann Hilbert problems in unknown generalized complex potential functions. These mathematical problems are then solved with the Riemann-Hilbert approach to obtain the stress and electric displacement components at any point of the domain. The explicit expressions for the outer saturated zone length, the crack opening potential(COP), the crack opening displacement(COD), and the local intensity factors(LIFs) are derived. A numerical study is presented for the modified strip saturated model in 2D arbitrary polarized semipermeable PZT-4 material. The obtained results are compared with those of the strip saturated model, and the effects of the polynomially varying saturation condition on the saturated zones and the applied electrical loadings are presented.     

Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

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     

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional(2 D) semipermeable piezoelectric media. In this model, an existing polarization saturation(PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement(COD), the crack opening potential(COP), and the local stress intensity factors(SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.     

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.     

Analytical modeling for nonlinear vibration analysis of partially cracked thin magneto-electro-elastic plate coupled with fluid

A nonlinear analytical model for the transverse vibration of cracked magneto-electro-elastic (MEE) thin plate is presented using the classical plate theory (CPT). The MEE plate material selected is fiber-reinforced \(\hbox {BaTiO}_{3}\)–\(\hbox {CoFe}_{2}\hbox {O}_{4}\) composite, which contains a partial crack at the center. The CPT and the simplified line spring model for crack terms are modified to accommodate the effect of electric and magnetic field rigidities. The analysis considers in-plane forces for the MEE plate, which makes the model nonlinear. The derived governing equation is solved by expressing the transverse displacement in terms of modal coordinates. An approximate solution for forced vibration of cracked MEE plate is also obtained using a perturbation technique. The effect of part-through crack, volume fraction of the composite on the vibration frequencies and structure response is investigated. The frequency response curves presented shows the phenomenon of hard or soft spring. Furthermore, the devised model is extended to the case of cracked MEE plate submerged in fluid. Velocity potential function and Bernoulli’s equation are used to incorporate the inertia effect of surrounding fluid. Both partially and totally submerged plate configurations are considered. The validation of the present results is carried out for intact submerged plate as to the best of the author’s knowledge the literature lacks in results for submerged-cracked plates. New results for cracked MEE plate show that the vibration characteristics are affected by volume fraction, crack length, fluid level and depth of immersion.     

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.     

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.     

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown(DB) model for a penny-shaped crack under a semipermeable boundary condition in a three-dimensional piezoelectric medium is studied.An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity,and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack.The effect of electric boundary conditions on crack faces is discussed on the basis of DB model.By comparing the DB model with the polarization saturation(PS) model for different piezoelectric materials,some interesting phenomena related to the electric yielding zone and local J-integral are observed.     

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the K _I -dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.     

Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding

A plane problem for two identical piezoelectric semi-infinite spaces adhered by means of a thin isotropic interlayer is considered. It is assumed that a crack of a limited electric permeability occurs in the interlayer parallel to its faces. Combined electromechanical loading is prescribed at infinity. It is assumed that the interlayer is softer than the adherent materials. To avoid the singularities, which are typical for the Griffith crack model, two distinct zones – a zone of mechanical yielding and a zone of electrical saturation – of unknown lengths are introduced as crack continuations. These lengths can be essentially different, with the zone of mechanical yielding significantly longer or shorter than the zone of electrical saturation. Assuming that the interlayer thickness tends to zero, a constant normal stress is prescribed in the zone of mechanical yielding and a saturated electrical displacement is prescribed in the zone of electrical saturation. Outside of these zones, the semi-infinite spaces are assumed to be perfectly bonded. This formulation results in a linear fracture mechanics problem with unknown pre-fracture zone lengths. The problem, formulated mathematically by a system of two equations of linear relationship, is solved exactly. The unknown yield and saturated zones lengths are found from the conditions of finiteness of stress and electrical displacement at the ends of these zones for the both cases when the electrical saturated zone is longer and shorter than the zone of mechanical yielding. It is shown that the same equation as for the Griffith crack model can be used for the determination of the electrical displacement in the crack region. The main results of the paper are obtained in the form of simple analytical equations which are convenient for engineering applications. Some numerical illustrations in graphical and tabular form show dependencies of the pre-fracture zone lengths, the energy release rate, the mechanical displacement and electrical potential jumps on the electromechanical loading and the electrical permeability of the crack medium.     

含力磁屈服区的压磁材料硬币型裂纹问题研究

本文利用Hankel 变换及Copson 求解方法,得到了在无穷远处磁荷载以及对称力荷载作用下的无限大压磁材料中,硬币型裂纹在裂纹尖端含有环状力磁双屈服区情况下的相关解析解．本文考虑了磁屈服区尺寸小于、大于、等于力屈服区尺寸的三种情况．结果表明：屈服区尺寸不仅与外荷载及材料常数相关,更与力磁屈服区尺寸的相对大小有关;其中,较大尺寸的屈服区仅受同性质单一荷载影响,与材料常数无关;而较小尺寸屈服区受力磁两荷载与材料常数共同影响;裂纹张开位移、磁势跳变均沿裂纹面径向增大而减小;裂尖张开位移、裂尖磁势跳变与材料常数相关,且随外荷载增大而增大．     

Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics

This paper presents a computational investigation of a proposed simplified account for electric displacement saturation on the hysteretic behavior of initially unpoled ferroelectric ceramics as well as on the initiation and propagation of cracks in poled ferroelectric ceramics within the linear regime of piezoelectricity. For the latter case, experimental observations suggest an odd dependency of the onset of crack initiation in these brittle materials on the orientation of the applied electric field with respect to their poling direction which contradicts theoretical results which propose an even dependency of the energy release rate on the applied electric field within the framework of anisotropic linear piezoelectricity. Electric non-linearities arising at regions of inhomogeneities such as inclusions or at the crack tip are proposed in the literature to avoid this discrepancy. Electric displacement saturation is one such non-linear effect which is investigated in this work. A simplified account of this effect is proposed based on an exponential saturation model of the identified material parameters which can be related to this non-linearity. Its advantage over the superposition of a complex function onto the singular solution of a crack within the framework of linear piezoelectricity lies in the straightforward extension of the proposed approach to problems where no analytical solutions exist. This is outlined based on its incorporation into a rate-dependent ferroelectric model accounting for polarization switching as well as based on its incorporation into a finite element framework capable of simulating the initiation and propagation of cracks in piezoelectric ceramics through strong discontinuities in the displacement field and the electric potential. It is shown that besides the determination of the crack initiation onset also the crack propagation direction is influenced by the appearance of saturation zones arising at the crack tip normal to the polarization direction. The numerically obtained crack paths are found to be close to the experimentally reported results.     

Fracture prediction for piezoelectric ceramics based on the electric field saturation concept

The electrical field saturation model is applied to the fracture prediction of piezoelectric materials containing electrically impermeable cracks. This model is analogously similar to the electric displacement saturation model that available in the literature. An electrical field saturation strip near the crack front is introduced in the analytical model. The stress intensity factor K and the energy release rate G are obtained in closed-form. It is found that fracture predictions based on K and G criteria are identical. Fracture predictions based on the electric field saturation model and the electric displacement model are also found to be the same.     

压电材料反平面III型裂纹电塑性区的分析

本文对反平面III型裂纹电塑性区进行了分析。采用条带模型得到了电塑性区大小的表达式。对于电塑性区的边界条件采用了两种处理方式,一是采用机械位移连续性边界条件,另一种是假设电塑性区的切应力保持为常数的假设。其中后一种处理方式消除了电场和应力在裂纹尖端的奇异性,与实际情况相符合。两种处理方式得到了相同的电塑性区的大小的表达式,并根据两种处理方式计算了能量释放率。类比Irwin的应力松弛模型,本文采用电位移松弛模型同样得到了电塑性区的大小。将条带模型得到的结果与电位移松弛模型得到的结果进行比较发现,在小范围塑性变形条件下,两种方法所得的结果比较接近,从而说明这两种方法的有效性,得到了比较满意的结果。