Exact solution for mixed boundary value problems at anisotropic piezoelectric bimaterial interface and unification of various interface defects

The special mixed boundary value problem in which a debonded conducting rigid line inclusion is embedded at the interface of two piezoelectric half planes is solved analytically by employing the 8-D Stroh formalism. Different from existing interface insulating crack model and interface conducting rigid line inclusion model, the presently analyzed model is based on the assumption that all of the physical quantities, i.e., tractions, displacements, normal component of electric displacements and electric potential, are discontinuous across the interface defect. Explicit solutions for stress singularities at the tips of debonded conducting rigid line inclusion are obtained. Closed form solutions for the distribution of tractions on the interface, surface opening displacements and jump in electric potential on the debonded inclusion are also obtained, in addition real form solutions for these physical quantities are derived. Various forms of interface defect problems encountered in practice are solved within a unified framework and the stress singularities induced by those interface defects are discussed in detail. Particularly, we find that the analysis of interface cracks between the embedded electrode layer and piezoelectric ceramics can also be carried out within the unified framework.     

Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites

We consider an arc-shaped conducting rigid line inclusion located at the interface between a circular piezoelectric inhomogeneity and an unbounded piezoelectric matrix subjected to remote uniform anti-plane shear stresses and in-plane electric fields. Moreover, one side of the rigid line inclusion has become fully debonded from the matrix or the inhomogeneity leading to the formation of an insulating crack. After the introduction of two sectionally holomorphic vector functions, the problem is reduced to a vector Riemann–Hilbert problem, which can be decoupled sequentially by repeated application of the orthogonality relations between the eigenvectors for two corresponding generalized eigenvalue problems.     

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.     

三相压电复合本构模型中的弧形界面裂纹

深入研究了三相同心圆柱压电复合本构模型中的弧形绝缘界面裂纹问题。采用复势方法获得了该问题的级数形式的解答,并给出了应力、应变、电位移和电场强度等物理量在全场及界面上的分布,同时推导了裂尖处广义强度因子及裂面张开位移和裂面上电势差的表达式。具体计算表明该级数解答收敛迅速,同时显示出第三相混杂区的影响是不能忽略的。由于裂尖处应力奇异性为－1／2,则这种解答不会出现平面应变状态下界面裂纹裂尖处的振荡奇异性,从而不会产生违反物理实际的裂面相互嵌入现象,则该弹性解答也是建立了坚实的物理基础之上。     

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroeiastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity , totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.     

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.     

ARC INTERFACE CRACK IN A THREEPHASE PIEZOELECTRIC COMPOSITE CONSTITUTIVE MODEL

An in-depth investigation is made on the problem of an arc-shaped interface insulating crack in a three-phase concentric circular cylindrical piezoelectric composite constitutive model. An exact solution in series form is derived by employing the complex variable method. In addition, the distribution of physical quantities such as stresses, strains, electric displacements and electric fields in the whole field and along the interface is also presented. Explicit expressions for crack opening displacement, jump in electric potential on the crack surface and the electro-elastic field intensity factors at the crack tips are obtained. Specific calculations demonstrate that the convergence of the series form solution is satisfactory and that the outer phase (composite phase) will exert a significant effect on the electro-mechanical coupling response of the composite system. Owing to the fact that stresses and electric displacements still possess conventional inverse square root singularities, the oscillating singularities near the crack tip under plane strain conditions will be absent and, as a result, no unphysical interpenetration phenomenon of the two crack surfaces will occur. In conclusion, the elastic solution obtained is also based on a solid physical foundation. Project supported by the National Natural Science Foundation of China (No.59635140), and the Doctorate Foundation of Xi'an Jiaotong University.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅰ)--SEVERAL EXACT SOLUTIONS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding exact solutions are obtained with the trial-anderror method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are rectangular beams having rigid body displacements and identical electrical potential, rectangular beams under uniform tension and electric displacement as well as pure shearing and pure bending, beams of two free ends subjected to uniform electrical potential on the upper and lower surfaces.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅰ)——SEVERAL EXACT SOLUTIONS

IntroductionDue to its excellent piezoelectric properties,composites made of piezoelectric materialsare found widespread applications and attracted more attentions[1-10].Because of materialanisotropy and couplingbetween mechanical deformation and electric…     

P波对界面部分脱胶的刚性圆柱夹杂物的散射

本文求解了弹性Ｐ波对界面部分脱胶的可动刚性圆柱夹杂物的散射问题。将脱胶区看作表面不相接触的弧形界面裂纹，借助波函数展开法并利用边界条件将问题转化为一组对偶级数方程。然后通过引入裂纹面的位错密度函数，将其化为一组具有Ｈｉｌｂｅｒｔ核的第二类奇异积分方程，并进一步化为Ｃａｕｃｈｙ型奇异积分方程组，数值求解方程组可获得动应力强度因子，夹杂物刚体振动位移和散射截面等重要参量。结果显示该类结构在较低的频率上发生共振，此低频共振特性与脱胶区大小，入射波方向、材料组合等多种参数有关。与已有方法相比，本文的方法更具一般性，适用于任意材料组合。     

An Interface Inclusion between Two Dissimilar Piezoelectric Materials

The generalized two-dimensional problem of a dielectric rigid line inclusion, at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity, is studied by means of the Stroh theory. The problem was reduced to a Hilbert problem, and then closed-form expressions were obtained, respectively, for the complex potentials in piezoelectric media, the electric field inside the inclusion and the tip fields near the inclusion. It is shown that in the media, all field variables near the inclusion-tip show square root singularity and oscillatory singularity, the intensity of which is dependent on the material constants and the strains at infinity. In addition, it is found that the electric field inside the inclusion is singular and oscillatory too, when approaching the inclusion-tips from inside the inclusion.     

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     

含椭圆形刚性夹杂的压电材料平面问题

应用复变函数的Ｆａｂｅｒ级数展开方法，本文研究了含椭圆形刚性夹杂的压电材料平面问题，给出了问题的封闭解。解签表明，夹杂内的电场强度和电位移为常量。并通过算例分析，讨论了正，逆压电效应在基体孔周处的机电行为。     

Analysis of a Circular Piezoelectric Semiconductor Embedded in a Piezoelectric Semiconductor Substrate

We analyze anti-plane electromechanical fields associated with a circular piezoelectric semiconductor of 6 mm symmetry embedded in a matrix of a different piezoelectric semiconductor. An exact solution is obtained. The solution shows the presence of field concentration near the interface. It is also found that the strain and electric fields inside the inclusion are not uniform.     

Basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials

The basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials was studied in this paper using the generalized Almansi’s theorem. To make the analysis tractable, it was assumed that the shear modulus varies exponentially along the horizontal axis parallel to the crack. The problem was formulated through a Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

含直对称裂纹的一维六方压电准晶对SH波的散射Scattering of SH wave by a linear symmetric crack at one-dimensional piezoelectric hexagonal quasicrystals

利用积分变换技术,结合Copson方法,研究了含直线型对称裂纹的一维六方压电准晶对SH波的散射问题。通过求解对偶积分方程,得到声子场、相位子场应力、位移及电场电位移分量的解析解。定义了裂纹尖端应力强度因子及电位移强度因子,给出了电非渗透性条件下应力强度因子及电位移强度因子的解析解。此研究结果对压电准晶材料的工程应用有一定的理论价值。     

A piezoelectric screw dislocation interacts with interfacial collinear rigid lines in piezoelectric bimaterials

Based on the complex variable method and perturbation technique, an analytical closed-form solution is derived for the interaction between a screw dislocation and collinear rigid lines along the interface of two dissimilar piezoelectric media under remote anti-plane mechanical and in-plane electrical loading. The rigid lines are either conducting or dielectric. The dislocation core is subjected to a line-force and a line-charge. A square-root singularity of field variables near the tip of an interfacial rigid line is observed. The rigid line extension force acting on the tip is obtained in terms of the strain and electric field intensity factor. The force on the dislocation due to the interfacial rigid line is calculated. The influence of the angular position of the dislocation, material properties and electromechanical coupling factor on the force is studied in detail.     

The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material

In this paper, the interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of the air inside the crack was considered. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the electric permittivity of the air inside the crack and the gradient parameter of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Dynamic solution of a multilayered orthotropic piezoelectric hollow cylinder for axisymmetric plane strain problems

The dynamic solution of a multilayered orthotropic piezoelectric infinite hollow cylinder in the state of axisymmetric plane strain is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state space method, and the dynamic part is obtained by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique. By using the obtained quasi-static and dynamic parts and the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered orthotropic piezoelectric infinite hollow cylinder consisting of arbitrary layers and subjected to arbitrary axisymmetric dynamic loads. Numerical results are finally presented and discussed.     

Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere

The dynamic solution of a multilayered spherically isotropic piezoelectric hollow sphere subjected to radial dynamic loads is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state-space method, and the dynamic part is obtained by the method of separation of variables coupled with the initial parameter method as well as the orthogonal expansion technique. By using the quasi-static and dynamic parts, the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered spherically isotropic piezoelectric hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric dynamic loads. Finally, numerical results are presented and discussed.