Asymptotic fields in the finite element analysis of electrically permeable interface cracks in piezoelectric bimaterials

Summary The interface crack problem for a piezoelectric bimaterial based on permeable conditions is studied numerically. 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 and electrical loads, the complex stress intensity factor for an interface crack is obtained. The influence of the applied loads on the electromechanical fields near the crack tip is also studied. For a particular case of a short crack with respect to the bimaterial size, the numerical results are compared with the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.One author (V.G.) gratefully acknowledges the support provided by the Alexander von Humboldt Foundation of Germany.accepted for publication 7 June 2004     

Interface crack-tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric bimaterials

Transversely isotropic piezoelectric (TIP) bimaterials with an impermeable interface crack have been classified [Int. J. Frac. 119 (2003) L41] into two classes corresponding to the vanishing of the two singularity parameters or κ. It is shown in the present paper that the related eigenvalue problems for either =0 or κ=0 are not degenerate. The crack-tip generalized stress fields are obtained subsequently. A new definition of crack-tip intensity factors is presented for interface cracks in practical TIP bimaterial of practical interest. Such defined intensity factors are real numbers, which dominate the maximum crack-tip stress singularity and do not generate any phase angle change under any dimension system transformation for physical quantities.     

On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric bimaterials

In this paper, characteristics of the interface crack-tip stress and electric displacement fields in transversely isotropic piezoelectric bimaterials are studied. The authors have proven, within the framework of the generalized Stroh formalism for piezoelectric bimaterials, that there is no coexistence of the parameters (oscillating) and κ (non-oscillating) in the interface crack-tip generalized stress field for all transversely isotropic piezoelectric bimaterials. This leads to the classification of piezoelectric bimaterials into one group that exhibits the oscillating property in the interface crack-tip generalized stress field and the other that does not. Fifteen (15) pair-combinations of six (6) piezoelectric materials PZT-4, PZT-5H, PZT-6B, PZT-7A, P-7, and BaTiO₃, which are commonly used in practice, are numerically analyzed in this study, and the results backup the above theoretical conclusions. Moreover, the associated eigenvectors for such material systems (with either =0 or κ=0) are also obtained numerically, and the result show that there still exist four linear independent associate eigenvectors for each bimaterial.     

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.     

Quasi-static and dynamic crack growth along bimaterial interfaces: A note on crack-tip field measurements using coherent gradient sensing

The paper presents a preliminary experimental investigation of crack-tip deformation fields near quasistatically and dynamically growing cracks in bimaterial interfaces. A three-point-bend bimaterial specimen with a relatively large stiffness mismatch between the two materials is studied. A recently developed optical method of coherent gradient sensing (CGS) is used to map crack-tip deformation fields.The quasi-static measurements are interpreted using plane-stress, singular-field solution for the interface crack tip. Results are compared with two-dimensional finite-element computations performed on identical specimen geometries and material mismatch.³² Impact studies conducted with bimaterial specimens provide the first experimental results of deformation fields near dynamically growing cracks along interfaces. Very high crack velocities, up to 80 percent of the Rayleigh wave speed for the less stiffer material of the two, are observed.     

压电圆柱形夹杂对邻近裂纹三维应力场的影响

研究了压电复合材料薄板中压电圆柱形夹杂与邻近宏观钝裂纹间的相互作用。重点分析了外加电场，裂尖与压电圆柱形夹杂间韧带长度对裂尖三维应力场的影响。计算结果表明：在不同的外加电场作用下，压电体不仅能改变裂尖张开应力的大小，还能改变其分布。所得结果对进一步探讨线弹性介质中裂纹的启裂控制有参考价值。     

有限尺寸下双材料界面附近裂纹位置对裂纹尖端的影响

随着复合材料的应用和发展,不同材料组成的界面结构越来越受到人们的重视.界面层两侧材料的性能相异会引起材料界面端奇异性,同时界面和界面附近存在裂纹会引起裂尖处的应力奇异性.因此双材料界面附近的力学分析是比较复杂的.论文建立双材料直角界面模型,在材料界面附近预设初始裂纹,计算了有限材料尺寸对界面应力场及其附近裂纹应力强度因子的影响.运用弹性力学中的Goursat公式求得直角界面端在有限尺寸下的应力场以及其应力强度系数.通过叠加原理和格林函数法进一步得到在直角界面端附近的裂纹尖端应力强度因子.计算结果表明,在适当范围内改变材料内裂纹与界面之间的距离,界面附近裂纹尖端的应力强度因子随着裂纹与界面距离的增加而减少,并且逐渐趋于稳定.分析结果可以为预测双材料结构复合材料界面失效位置提供参考.     

An interface crack with partially electrically conductive crack faces under antiplane mechanical and in-plane electric loadings

An interface crack in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the crack faces is electrically conductive while the other part is electrically permeable. All electro-mechanical values are presented using sectionally-analytic vector-functions and a combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Simple analytical expressions for the shear stress, electric field and also for mechanical displacement jump of the crack faces are derived. These values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear stress, electric field and electric displacement jump are found. Their intensity factors are determined as well. Intensity factors variations with respect to the external electric field and different ratios between the electrically conductive and electrically permeable crack face zones are also demonstrated.     

扩展裂纹尖端的塑性热耗散与温度场

材料的不可逆变形功以热的形式耗散，形成温度场，本文考虑Ⅰ型裂纹尖端过程区塑性变形功的热耗散，视裂尖塑性过程区为内热源，通过合理地构造一个热源密度函数，结合裂尖塑性区的近似模型，给出了裂纹定常扩展过程中的裂尖温度场。     

双材料界面裂纹复应力强度因子的正则化边界元法

双材料界面裂纹渐近位移和应力场表现出剧烈的振荡特性,许多用于表征经典平方根(r1/2)和负平方根(r?1/2)渐近物理场的传统数值方法失效,给界面裂纹复应力强度因子(K1+iK2)的精确求解增加了难度.引入一种含有复振荡因子的新型"特殊裂尖单元",可精确表征裂纹尖端渐近位移和应力场的振荡特性,在避免裂尖区域高密度网格剖...     

Interaction of a conductive crack and of an electrode at a piezoelectric bimaterial interface

The interaction of a conductive crack and an electrode at a piezoelectric bi-material interface is studied. The bimaterial is subjected to an in-plane electrical field parallel to the interface and an anti-plane mechanical loading. The problem is formulated and reduced, via the application of sectionally analytic vector functions, to a combined Dirichlet–Riemann boundary value problem. Simple analytical expressions for the stress, the electric field, and their intensity factors as well as for the crack faces' displacement jump are derived. Our numerical results illustrate the proposed approach and permit to draw some conclusions on the crack–electrode interaction.     

Field structure at mode III dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Elastic-plastic crack growth on plane strain bimaterial interface

Finite element computation are carried out to simulate plane strain crack growth on a bimaterial interface under the assumption of small scale yielding. The modified Gurson constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. It is found from the calculation that the critical fracture toughness for crack growth is much lower in bimaterials than that in homogeneous material. The critical fracture toughness is strongly dependent on material properties of the bimaterial pair and the mixed mode of remote loads. The interface crack grows in the more compliant (lower hardening) material or in the weaker (lower yield strength) material. In Mode-I loading, the crack grows zigzag along the interface. Project supported by Fok Ying-Tung Education Foundation and National Natural Science Foundation of China.     

ASYMPTOTIC ANALYSIS OF MODE Ⅱ STATIONARY GROWTHCRACK ON ELASTIC-ELASTIC POWER LAWCREEPING BIMATERIAL INTERFACE

A mechanical model was established for mode Ⅱ interfacial crack static growingalong an elastic-elastic power law creeping bimaterial interface. For two kinds of boundaryconditions on crack faces, traction free and frictional contact, asymptotic solutions of thestress and strain near tip-crack were given. Results derived indicate that the stress andstrain have the same singularity, there is not the oscillatory singularity in the field; thecreep power-hardening index n and the ratio of Young‘s module notably influence the crack-tip field in region of elastic power law creeping material and n only influences distribution ofstresses and strains in region of elastic material. When n is bigger, the creepingdeformation is dominant and stress fields become steady, which does not change with n.Poisson‘s ratio does not affect the distributing of the crack-tip field.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

双材料界面裂纹的改进广义有限差分法

采用数值方法进行断裂力学分析时,裂纹尖端奇异区域处理的好坏直接关系到最终断裂力学参数的求解精度。与传统均匀介质不同,复合材料界面裂纹渐近位移和应力场表现出剧烈的振荡特性,许多用于表征经典的平方根和负平方根物理场渐近性的传统方法也因此失效。论文提出了一种改进的广义有限差分法,该方法基于多元函数泰勒级数展开和移动最小二乘法的思想,将节点变量的各阶导数由相邻点集函数的加权线性累加来近似,具有无网格、无数值积分、数据准备简单、稀疏矩阵快速求解等优点。为提高该方法求解断裂力学问题的计算精度和数值稳定性,论文引入了裂尖奇异区域局部点簇的自动创建技术和一种基于局部点簇几何尺寸的矩阵正则化算法。数值算例表明,所提算法稳定,效率高,在不增加计算量的前提下,显著提高了裂尖近场力学参量和断裂力学参数的求解精度和数值稳定性。     

Scattering of SH-waves by an interacting interface linear crack and a circular cavity near bimaterial interface

An analytical method is developed for scattering of SH-waves and dynamic stress concentration by an interacting interface crack and a circular cavity near bimaterial interface. A suitable Green‘s function is contructed, which is the fundamental solution of the displacement field for an elastic half space with a circular cavity impacted by an out-plane harmonic line source loading at the horizontal surface. First, the bimaterial media is divided into two parts along the horizontal interface, one is an elastic half space with a circular cavity and the other is a complete half space. Then the problem is solved according to the procedure of combination and by the Green‘s function method. The horizontal surfaces of the two half spaces are loaded with undetermined anti-plane forces in order to satisfy continuity conditions at the linking section, or with some forces to recover cracks by means of crack-division technique. A series of Fredholm integral equations of first kind for determining the unknown forces can be set up through continuity conditions as expressed in terms of the Green‘s function. Moreover, some expressions are given in this paper, such as dynamic stress intensity factor (DSIF) at the tip of the interface crack and dynamic stress concentration factor (DSCF) around the circular cavity edge. Numerical examples are provided to show the influences of the wave numbers, the geometrical location of the interface crack and the circular cavity, and parameter combinations of different media upon DSIF and DSCF.