Reliability assessment of a bi-material notch: Strain energy density factor approach

Joints of different materials have many applications in structural engineering and microelectronics. In the present contribution the joint is modelled as a bi-material notch. The singular stress field near the notch tip is investigated. Depending on the notch geometry and materials, the stress field can have one or two singularities. It is shown that to study the problem of a crack onset at the notch, both terms have to be taken into account. Criteria for the direction and for crack nucleation are formulated. The approach utilizes the knowledge of the strain energy density factor distribution in a bi-material notch vicinity.     

压电材料中裂纹面电边界条件的适用性

先分析理想裂纹面电边界条件(包括导通型裂纹和不导通型裂纹),然后分析缺陷的厚度及其对尖端场的影响,用封闭形式的解表示出了缺陷厚度的影响．结果指出,裂纹长厚比对计算结果是有影响的,除非裂纹极扁,不可通裂纹边界条件可以给出合理的结果,而可通边界条件只有在一些限制条件下才适用。     

APPLICABILITY OF THE CRACK FACE ELECTRICAL BOUNDARY CONDITIONS IN PIEZOELECTRIC MECHANICS

The electrical boundary conditions on the crack faces and their applicability in piezoelectric materials are discussed. A slit crack and a notch of ?nite thickness in piezoelectric materials subjected to combined mechanical and electrical loads is consi…     

PZT-4紧凑拉伸试样的断裂分析

基于线性压电材料的复势理论,通过解析分析,导出了一种分析有限压电板裂纹问题的解析数值方法. 首先,计算了含中心裂纹有限板的断裂参数,与Woo和Wang的解析数值法(Int J Fract, 1993, 62: 203$\sim$218)相比较,表明该方法具有很高的精度和很好的计算效率. 随后,采用该方法和有限元法计算了PZT-4紧凑拉伸试样在绝缘裂纹面边界条件下断裂时的断裂参数,发现各断裂参数的临界值分散性很大,不能作为压电材料的单参数断裂准则. 进而,针对试样真实的裂隙形状,采用有限元法计算了裂隙尖端的应力、电位移场,比较了裂隙内介质的介电性能对裂隙尖端场的影响,计算了带微裂纹的真实裂隙模型的断裂参数并进行了理论分析.      

Interaction of a screw dislocation with a notch in a piezoelectric bi-material

Summary The electro-elastic interaction of a screw dislocation and a notch in a piezoelectric bi-material is analyzed. The electro-elastic fields induced by the dislocation are derived using the conformal mapping and the image-dislocation approach, where the solution for a piezoelectric bi-material without a notch is used as a base. The stress and the electric displacement intensity factors of the notch and the image force on the dislocation are given explicitly. We find that intensity factors are expressed in terms of the effective material constants, while the radial component of the image force is independent of the notch angle and the angular position of the dislocation in the polar coordinate system. Numerical results for the image force are provided for the use when one of the two media is purely elastic. They illustrate the behavior of the dislocation in the neighborhood of the notch.     

Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞. Biography: LI Xian-fang (1964-)     

Applicability of the crack-face electromagnetic boundary conditions for fracture of magnetoelectroelastic materials

This paper discusses the different electromagnetic boundary conditions on the crack-faces in magnetoelectroelastic materials, which possess coupled piezoelectric, piezomagnetic and magnetoelectric effects. A notch of finite thickness in these materials containing air (or vacuum) is also addressed. Four ideal crack-face electromagnetic boundary condition assumptions, that is, (a) electrically and magnetically impermeable crack, (b) electrically impermeable and magnetically permeable crack, (c) electrically permeable and magnetically impermeable crack and (d) electrically and magnetically permeable crack, are investigated separately. The influence of notch thickness on the field intensity factors at notch tips and the electromagnetic field inside the notch are obtained in closed-form. The results are compared with the ideal crack solutions. Applicability of crack-face electromagnetic boundary condition assumptions is discussed.     

The interaction between crack and electric dipole of piezoelectricity

Discrete dipoles located near the crack tip play an important role in nonlinear electric field induced fracture of piezoelectric ceramics. A physico-mathematical model of dipole is constructed of two generalized concentrated piezoelectric forces with equal density and opposite sign. The interaction between crack and electric dipole in piezoelectricity is analyzed. The closed form solutions, including those for stress and electric displacement, crack opening displacement and electric potential, are obtained. The function of piezoelectric anisotropic direction,p _a (θ)=cosθ+p _a sinθ, can be used to express the influence of a dipole's direction. In the case that a dipole locates near crack tip, the piezoelectric stress intensity factor is a power function with −3/2 index of the distance between dipole and crack tip. Supported by National Natural Science Foundation of China(No. 10072033)     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程，并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

夹杂对两相材料界面裂纹的影响

根据界面上应力和位移的连续条件,得到了单向拉伸状态下,含有椭圆夹杂的无限大双材料组合板的复势解。进一步通过求解Hilbert问题,得到了含有夹杂和半无限界面裂纹的无限大板的应力场,并由此给出了裂尖的应力强度因子K。计算了夹杂的形状、夹杂的位置、夹杂的材料选取以及上、下半平面材料与夹杂材料的不同组合对裂尖应力强度的影响。计算结果表明夹杂到裂尖的距离和夹杂材料的性质对K影响较大,对于不同材料组合,该影响有较大差异。夹杂距裂尖较近时,会对K产生明显屏蔽作用,随着夹杂远离裂尖,对K的影响也逐渐减小。另外,软夹杂对K有屏蔽作用,硬夹杂对K有反屏蔽作用,而夹杂形状对K几乎没有影响。     

压电效应下一维六方准晶双材料孔边裂纹的反平面问题研究

利用复变函数方法和保角变换技术研究了压电效应下一维六方准晶双材料中圆孔边单裂纹的反平面问题．考虑电不可渗透型边界条件,运用保角变换和Stroh公式得到了弹性体受远场剪切力和面内电载荷作用下裂纹尖端应力强度因子和能量释放率的解析解. 数值算例分析了几何参数、远场受力、电位移载荷对能量释放率的影响．结果表明：裂纹长度、耦合系数和远场剪切力的减小可以抑制裂纹的扩展．不考虑电场时,声子场应力对能量释放率的影响较小．本文的研究结果可作为研究一维六方压电准晶双材料孔边裂纹问题的理论基础,同时为压电准晶及其复合材料的设计、制备、优化和性能评估提供理论依据．     

Numerical simulation of loading edge cracks by edge impact using the extended finite element method

The extended finite element method is used to analyze a plate with two parallel edge cracks impacted by a cylindrical projectile. The influence of the impact speed, crack length,plate thickness and notch tip radius on the crack initiation and propagation is studied. Dynamics equations are solved by an implicit time integration scheme which is unconditionally stable. Very good agreement is achieved between numerical predictions and experimental results. The critical velocity of the crack initiation under different conditions is examined. The influence of the crack length is greater than that of the impact speed, plate thickness and notch tip radius.     

METHOD TO CALCULATE STRESS INTENSITY FACTOR OF V-NOTCH IN BI-MATERIALS

Based on Zak＇s stress function, the eigen-equation of stress singularity ofbi-materials with a V-notch was obtained. A new definition of stress intensity factor for a perpendicular interfacial V-notch of bi-material was put forward. The effects of shear modulus and Poisson＇s ratio of the matrix material and attaching material on eigen-values were analyzed. A generalized expression for calculating/（i of the perpendicular V-notch of bi-materials was obtained by means of stress extrapolation. Effects of notch depth, notch angle and Poisson＇s ratio of materials on the singular stress field near the tip of the V-notch were analyzed systematically with numerical simulations. As an example, a finite plate with double edge notches under uniaxial uniform tension was calculated by the method presented and the influence of the notch angle and Poisson＇s ratio on the stress singularity near the tip of notch was obtained.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

Anti-plane shear crack normal to and terminating at the interface of two bonded piezoelectric ceramics

The electroelastic analysis of two bonded dissimilar piezoelectric ceramics with a crack perpendicular to and terminating at the interface is made. By using Fourier integral transform, the associated boundary value problem is reduced to a singular integral equation with generalized Cauchy kernel, the solution of which is given in closed form. Results are presented for a permeable crack under anti-plane shear loading and in-plane electric loading. Obtained results indicate that the electroelastic field near the crack tip in the homogeneous piezoelectric ceramic is dominated by a traditional inverse square-root singularity, while the electroelastic field near the crack tip at the interface exhibits the singularity of power law r^−α, r being distance from the interface crack tip and α depending on the material constants of a bi-piezoceramic. In particular, electric field has no singularity at the crack tip in a homogeneous solid, whereas it is singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order and field intensity factors.     

Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids

Crack tip fields are calculated under plane strain small scale yielding conditions. The material is characterized by a finite strain elastic–viscoplastic constitutive relation with various hardening–softening–hardening hardness functions. Both plastically compressible and plastically incompressible solids are considered. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The initial crack is taken to be a semi-circular notch and symmetry about the crack plane is imposed. Plastic compressibility is found to give an increased crack opening displacement for a given value of the applied loading. The plastic zone size and shape are found to depend on the plastic compressibility, but not much on whether material softening occurs near the crack tip.On the other hand, the near crack tip stress and deformation fields depend sensitively on whether or not material softening occurs. The combination of plastic compressibility and softening(or softening–hardening) has a particularly strong effect on the near crack tip stress and deformation fields.     

Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading

A new experimental technique for accelerated fatigue crack growth tests was recently developed (Du et al., 2001). The technique, which uses piezoelectric actuators, enables application of cyclic loading at frequencies several orders higher than that by mechanical loading. However, the validity of this technique relies on the equivalence between piezoelectric and mechanical loading. In this paper, the behavior of an interfacial crack between a piezoelectric material and an elastic material under in-plane electric loading is studied. The displacement mismatch along a bonded interface due to electric potential loading on the piezoelectric material is modeled by inserting an array of uniformly distributed dislocations along the interface. By means of Fourier transformation methods, the governing equations are converted to an integral equation, which is then converted to a standard Hilbert problem. A closed form solution for stresses, electric field, and electric displacements along the bonded interface is obtained. The results agree very well with those obtained from numerical simulations. The results show that the closed form solution is accurate not only for far field distributions of stresses and electric variables, but also for the asymptotic distributions near the crack tip. The solution also suggests the likelihood of domain switching in the piezoelectric material near the crack tip, a process that may influence the interfacial fracture resistance.     

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh’s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials: Epoxy, Polymer, Al₂O₃, SiC, and Si₃N₄ and seven kinds of practical piezoelectric ceramics: PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities: and , which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials. The project supported by the National Natural Science Foundation of China (10572110), Doctor Foundation of the Chinese Education Ministry and Doctorate Foundation of Xi’an Jiaotong University. The English text was polished by Yunming Chen.     

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.     

A crack in a piezoelectric layer bonded to a dissimilar elastic layer under transient load

Summary  A piezoelectric layer bonded to the surface of an elastic structure is considered. The piezoelectric and the elastic layers are infinite along the x-axis and have finite thickness in the y-direction. The polarization direction of the piezoelectric material is along the y-axis. By means of the method of singular integral equations, the solution in a Laplace transform plane is demonstrated. Laplace inversion yields the results in the time domain. Numerical values of the crack tip fields under in-plane transient electromechanical loading are obtained. The influence of layers thickness on stress and electric displacement intensity factors is investigated. Received 16 March 2000; accepted for publication 16 August 2000