Crack deflection at an interface between dissimilar piezoelectric materials

The problem of crack deflection in bimaterial systems is considered in this paper. The material combinations may be of piezoelectric-piezoelectric, or one is piezoelectric and the other is not. Based on the Stroh formulation for anisotropic material, Green's functions for various bimaterial combinations are presented within the framework of two-dimensional electroelasticity, allowing the crack problem to be expressed in terms of coupled singular integral equations. A crack impinging on an interface joining two dissimilar materials may arrest or may advance be either penetrating the interface or deflecting into the interface. The competition between deflection and penetration is investigated using the maximum energy release rate criterion. Numerical results are presented to study the role of remote electroelastic loads on the path selection of crack extension. Key words: Crack, piezoelectric material, interface, Green's function, singular integral equation.     

Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials

Interface and interfacial cracks interacting with screw dislocations in piezoelectric bimaterials subjected to antiplane mechanical and in-plane electrical loadings are studied within the framework of linear piezoelectricity theory. Straight dislocations with the Burgers vector normal to the isotropic basal plane near the interface or interfacial crack are considered. The dislocations are characterized by a discontinuous electric potential across the slip plane and are subjected to a line-force and a line-charge at the core. An explicit solution for the screw dislocation in piezoelectric bimaterial with straight interface is found based on the solution of a similar problem for infinite homogenous medium. The obtained relation is independent of the nature of singularity. This fundamental result is used to analyze dislocation interacting with a set of collinear interfacial cracks in piezoelectric bimaterials. Three solutions for the screw dislocation interacting with a semi-infinite crack, finite crack, and edge crack between two bonded dissimilar piezoelectric materials are obtained in closed-form. These solutions can be used as Green’s functions for the analyses of interfacial cracks in piezoelectric bimaterials.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interface crack problem in elastic dielectric/piezoelectric bimaterials

By considering an isotropic elastic dielectric material as a transversely isotropic piezoelectric material with little piezoelectricity, the interface crack problem in elastic/piezoelectric bimaterials is treated in this paper based on Stroh's complex potential theory (1958) with the impermeable crack model. In order to obtain universal results, Numerical results of the near tip stress field and the electric field for 35 kinds of dissimilar bimaterials constructed by five kinds of elastic dielectric materials, namely Epoxy, Polymer, Al₂O₃, SiC and Si₃N₄, and seven kinds of piezoelectric ceramics, namely PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC151, are presented. It is concluded that all the combinations lead to the same results: in which the first crack tip singularity parameter does not vanish whereas the second parameter always vanishes. From the physical point of view, an interface crack in such a dissimilar material shows a similar oscillating singularity as that in dissimilar elastic bimaterials. Moreover, by using a maximization value technique, the regular inverse square root singularity r ^–1/2 of the stress and the electric field at the crack tip can be realized, although, theoretically, an interface crack in such bimaterials possesses the well-known oscillating singularity r ^{–1/2± i}. Of great significance is that, in the absence of mechanical loadings, a purely electric loading can induce relative large model I or II stress intensity factor for a interface crack in some elastic/piezoelectric bimaterials, which implies that the electric-induced failure may be realized in such bimaterials.     

The Coulombic traction on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials

This paper deals with the Coulombic traction usually neglected, but inherently acting, on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials. The dielectric material phase is treated as a special kind of piezoelectric material with a little piezoelectricity, whereas the metal phase is treated as another special kind of piezoelectric material with an extremely large permittivity and an extremely small piezoelectricity. The permittivity of the medium inside the crack gap is accounted for either. The normal electric displacement component and the Coulombic traction on the crack surfaces are unknown, and are determined from a cubic equation deduced from the extended Stroh formula. Numerical results for the Coulombic traction in both kinds of bimaterials reveal that in most cases its magnitude is remarkable and cannot be entirely neglected when the applied electric field is higher. It is concluded that in most cases the Coulombic traction yields significant influence on the effective stress intensity factor at the crack tip and may influence the fracture behavior in such kinds of bimaterials. As compared to homogenous piezoelectric materials, the metal phase always decreases the Coulombic traction, whereas the dielectric phase decreases it under the negative electric field and increases it under the positive electric field. In all cases, BaTiO₃ always yields a much larger Coulombic traction than PZT-4.     

Parallel crack near the interface of magnetoelectroelastic bimaterials

A parallel crack near the interface of magnetoelectroelastic bimaterials is considered. The crack is modelled by using the continuously distributed edge dislocations, which are described by the density functions defined on the crack line. With the aid of the fundamental solution for the edge dislocation, the present problem is reduced to a system of singular integral equations, which can be numerically solved by using the Chebyshev numerical integration technique. Then, the stress intensity factor (SIF), the magnetic induction intensity factor (MIIF) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Using these fracture criteria, the cracking behaviour of magnetoelectroelastic bimaterials is investigated. Numerical examples demonstrate that the interface, mechanical load, magnetic load and material mismatch condition are all important factors affecting the fracture toughness of the magnetoelectroelastic bimaterials.     

双相介质中椭圆孔洞与裂纹对SH波的散射

采用Green函数法和保角映射法解答了双相介质界面附近一个椭圆孔洞和一个裂纹(在同一侧)对SH波的散射问题。沿水平界面将双相介质剖分为一个含椭圆孔和裂纹的半空间以及一个完整的弹性半空间。结合“裂纹切割”法,利用Green函数法构造裂纹,求解出孔洞与裂纹同时存在时的位移和应力表达式。一组未知力系施加在水平界面上,使两部分契合,基于界面连续条件推导出一系列Fredholm积分方程组,从而求出未知力系。最后,给出算例讨论了不同参数对椭圆孔周边动应力集中系数和裂纹尖端动应力强度因子的影响。     

双相介质界面附近裂纹的断裂力学特征

复合材料界面附近的力学性态对于材料的性能和强韧化影响是非常重要的。首先研究和讨论了含裂纹的双相介质的J 积分守恒定律的适用性问题, 采用有限元法证明了当裂纹平行靠近界面时, 其J 积分数值与裂纹位置无关的假设。文中建立了一种双相介质界面附近存在斜裂纹的分析模型, 用有限元和数值拟合相结合的方法, 得到了在远场单轴拉应力作用下, 斜裂纹处在不同介质中, 近界面一端裂尖的é 型能量释放率近似计算公式, 和相应的应力强度因子的计算方法。     

加层压电条界面裂纹的稳态扩展

研究当压电条同时与两个不同材料的弹性条粘接在一起,在反平面机械载荷及面内电载荷联合作用下,长度不变的有限Griffith 界面裂纹沿加层压电条界面以常速稳态扩展时裂尖的动态断裂问题.应用Fourier积分变换将问题化为以第二类Fredholm积分方程表示的对偶积分方程,导出了相应的动应力强度因子表达式.给出了动应力强度因子与裂纹传播速度、裂纹长度、压电条及弹性条厚度、电荷载大小及方向的关系曲线.研究结果对结构设计及结构失效的预防具有理论和应用价值.     

Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system

A penny-shaped crack at the interface of a piezoelectric bi-material system is considered. Analytical general solutions based on Hankel integral transforms are used to formulate the mixed-boundary value problem corresponding to an interfacial crack and the problem is reduced to a system of singular integral equations. The integral equations are further reduced to two systems of algebraic equations with the aid of Jacobi polynomials and Chebyshev polynomials. Thereafter, the exact expressions for the stress intensity factors and the electric displacement intensity factor at the tip of a crack are obtained. Selected numerical results are presented for various bi-material systems to portray the significant features of crack tip fracture parameters and their dependence on material properties, poling orientation and electric loading.     

Dynamics response of complex defects near bimaterials interface by incident out-plane waves

Dynamics response of an elliptical cavity and a crack (on different sides) near bimaterials interface under incident out-plane waves is studied by applying the methods of complex variables and Green’s function. Firstly, based on “conjunction,” the analytical model is divided along the horizontal interface into an elastic half-plane possessing an elliptical cavity and a full elastic half-plane containing a crack. Using complex variables, the scattering displacement field of the half-plane containing an elliptical cavity under incident out-plane waves is then derived. According to the method of Green’s function, the corresponding Green’s functions of two half-planes impacted by an out-plane source load are further deduced. Combined with “crack division,” a crack at the full elastic the half-plane is created, and thus, expressions of displacement and stress are derived while the cavity coexists with the crack. Undetermined antiplane forces are loaded on the horizontal surfaces for conjunction of two sections and then solved by a series of Fredholm integral equations on account of continuity conditions of the interface. Finally, this paper focuses on the discussion of the influence law of different parameters on the dynamics response of complex defects near bimaterials interface by comprehensive numerical results.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Contact zone models for an interface crack in a piezoelectric material

Summary A plane strain problem for an interface crack along the fixed edge of a piezoelectric semi-infinite space is examined. Electrically conducting and electrically insulated crack surfaces are considered. By using Fourier transforms the systems of singular integral equations are formulated for both cases. It was found that in the second case for the most commonly used piezoelectric materials instead of oscillating singularity the real singularity of general power type occurs. The dependence of this singularity on the piezoelectric parameters has been investigated. The contact zone model is considered as an alternative one for the case of the oscillating singularity, and the way this model can be used for the investigation of interface cracks in finite size piezoelectrics is suggested.     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

The J-integral at Dugdale cracks perpendicular to interfaces of materials with dissimilar yield stresses

The J-integral is applied to a Dugdale crack perpendicular to an interface of materials with equal elastic properties but different yield stresses. It is shown that the integral is path independend with certain limitations to the integration path. Three essentially different paths can be distinguished. The first integration path is totally within the first material, it provides the local crack driving force. Performing the integral around the plastic zone in both materials gives the global crack driving force. An interface force can be defined by evaluating the integral along both sides of the plastically deformed region of the interface. A comparison of these three integrals reveals that the global crack driving force is equal to the sum of the local crack driving force and of the interface force. The derived expression for the J-integral are compared with the crack tip opening displacement published recently. This reveals that the local J describes the plastic deformation at the crack tip. Therefore it represents the crack driving force in bimaterials as it does the conventional J-integral in case of homogeneous materials. The analyses are also extended to cyclic plasticity, where an out-of-phase effect is observed. Finally it is discussed how these results can be used to explain fatigue tests at bimaterial specimens.     

Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial half-space

The problem of an edge-bridged crack terminating perpendicular to a bimaterial interface in a half-space is analyzed for a general case of elastic anisotropic bimaterials and specialized for the case of orthotropic bimaterials. The edge crack lies in the surface layer of thickness h bonded to semi-infinite substrate. It is assumed that long fibres bridge the crack. Bridging model follows from the assumption of “large” slip lengths adjacent to the crack faces and neglect of initial stresses. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. With respect to the bridged crack problems in finite dissimilar bodies, the reciprocal theorem (Ψ-integral) is demonstrated as to compute, in the present context, the generalized stress intensity factor through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. Also the application of the configurational force mechanics is discussed within the context of the investigated problem.     

Asymptotic stresses around a crack tip at the interface between planar or cylindrical bodies

Closed-form equations are derived for the asymptotic stresses in the neighborhood of a crack tip impinging on an interface between two isotropic materials. The symmetric problem is considered and follows from an exact elasticity solution formulated by Gupta [1]. The equations are valid for the planar problem, where the interface is straight and also for an axisymmetric problem in the presence of an annular or penny-shaped crack. The equations may serve to establish a tentative criterion that defines the subsequent direction of a crack impinging on a bimaterial interface. The ambiguity of the asymptotic stress state is highlighted and plausible application of the results is discussed.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Full-field analysis and image forces for piezoelectric bimaterials

Summary In this study, the two-dimensional explicit full-field solutions of transversely isotropic piezoelectric bimaterials subjected to mechanical and electrical loads are derived by using the Fourier-transform technique. The major objective of this study is to analyze the physical meaning and the structure of the solution. One of the novel features is that Green’s functions for bimaterials consist of Green’s functions for the infinite plane. The complete solutions of this problem include Green’s function of originally applied singularities in an infinite medium and nine image singularities which are induced to satisfy interface continuity conditions. It is shown that the physical meaning of the solution is the image method. The mathematical method used in this study provides an automatic determination for the locations of image singularities. The locations of image singularities are dependent on the roots of the characteristic equation for bimaterials. According to the characteristic roots, the number and distribution for image singularities are discussed in detail. The expressions for image forces acting on edge dislocations are given explicitly with the aid of the generalized Peach–Koehler formula. Numerical results for the full-field distributions of stresses, electric fields in bimaterials and image forces for edge dislocations are presented. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.