The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

The surface crack problem for a layered plate with a functionally graded nonhomogeneous interface

The plane elasticity solution is presented in this paper for the crack problem of a layered plate. A functionally graded interfacial region is assumed to exist as a distinct nonhomogeneous transitional layer with the exponentially varying elastic property between the dissimilar homogeneous surface layer and the substrate. The surface layer contains a crack perpendicular to the boundaries. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The main results presented are the variations of stress intensity factors as functions of geometric and material parameters of the layered plate.     

Energy release rate and bifurcation angles of piezoelectric materials with antiplane moving crack

The dynamic fracture problems of the piezoelectric materials with antiplane moving crack are analysed by using function of complex variable in the paper. The results show that the coupled elastic and electric fields inside piezoelectric media depend on the speed of the crack propagation, and have singularity at the crack tip. The stress intensity factor is independent of the speed of the crack propagation, which is identical to the conclusion of purely elasticity. Moreover, independent of the electric loading, the dynamic energy release rate can be expressed by the stress intensity factor and enlarge with the increase of crack speed. High speed of the crack moving could impede the crack growth. At the same time, the crack can be propagated into either curve or bifurcation if the crack speed is higher than the critical speed.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

Griffith crack moving along the interface of two dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

On contact zone models for an electrically impermeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x ₂ the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.     

Thermal stress intensities at an interface crack between two elastic layers

The thermal stress intensities (energy release rate and stress intensity factors) due to temperature changes are derived in closed-form for an interface crack between two elastic layers of dissimilar materials. The solutions are two-dimensional and tabulated over a wide range of material and layer thickness combinations. The tables serve as rapid evaluations of the thermal stress intensities for given temperature changes. A strain gauge technique is given for determining constraint coefficients which reflect the constraint conditions during the temperature changes. The solutions are compared with results from the literature. The stress intensities due to thermal and mechanical loads are generally superimposed. As an example of application, the solutions are utilized to obtain the complete thermal and mechanical stress intensities for a four-point bend specimen.     

Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.     

Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials

Weight functions proposed for interface cracks in dissimilar isotropic materials (Gao, 1991; Chen and Hasebe, 1994) are extended to treat those in piezoelectric materials. The difficulties in separating the eight distinct complex arguments are overcome. The pseudo-orthogonal properties of the eigenfunction expansion form found in isotropic dissimilar cases(Chen and Hasebe, 1994) are proved to be valid in the present cases although the mathematical manipulations performed here seem much more complicated than those in isotropic dissimilar materials. Several path-independent integrals are obtained and all the coefficients in the eigenfunction expansion form, including the K _I, K _II, K _III and K _e, could be calculated by the weight functions introduced in this paper. It is concluded that the weight functions presented here provide a powerful tool to calculate the dominant parameters at the interface crack tip without any special treatment to the singular stress field of the near-tip region.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。     

Transient response of a Griffith crack between dissimilar piezoelectric layers under anti-plane mechanical and in-plane electrical impacts

The problem of an interface crack between dissimilar piezoelectric layers under mechanical and electrical impacts is formulated by using integral transform and Cauchy singular integral equation methods. The dynamic stress intensity factor and dynamic energy release rate (DERR) are determined through use of the obtained solutions and the effects of the loading ratio, the geometry of crack configuration and the combination of material parameters on the above two quantities are discussed. The numerical calculations indicate that the electrical load can promote or retard the crack growth depending on its magnitude, direction and the existence of the mechanical load and that with the increase of the value of ratio of two material parameters, some material parameters will inhibit the crack growth. On the other hand, some material parameters play the contrary roles. In addition, the geometry of the crack configuration has the significant effects on the DERR. Finally the results are compared with those obtained in a previous investigation.     

Interaction of a penny-shaped crack with an elliptic crack under shear loading

The problem of interaction between equal coplanar elliptic cracks embedded in a homogeneous isotropic elastic medium and subjected to shear loading was solved analytically by Saha et al. (1999) International Journal of Solids and Structures 36, 619–637, using an integral equation method. In the present study the same integral equation method has been used to solve the title problem. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter _i up to the order _i⁵,(i=1,2) for both the elliptic as well as penny-shaped crack. Expressions for modes II and III stress intensity factors have been given for both the cracks. The present solution may be treated as benchmark to solutions of similar problems obtained by various numerical methods developed recently. The analytical results may be used to obtain solutions for interaction between macro elliptic crack and micro penny-shaped crack or vice-versa when the cracks are subjected to shear loading and are not too close. Numerical results of the stress-intensity magnification factor has been illustrated graphically for different aspect ratios, crack sizes, crack separations, Poisson ratios and loading angles. Also the present results have been compared with the existing results of Kachanov and Laures (1989) International Journal of Fracture 41, 289–313, for equal penny-shaped cracks and illustrations have been given also for the special case of interaction between unequal penny-shaped cracks subjected to uniform shear loading.     

Edge cracked piezoelectric ceramic block under electromechanical impact loading

The dynamic field intensity factors and energy release rates in a piezoelectric ceramic block containing an edge crack with the condition of continuous electric crack faces under electromechanical impact loading are obtained. Integral transform method is used to reduce the problem to two pairs of dual integral equations, which are then expressed to an Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor and dynamic energy release rate are obtained to show the influence of the geometry and electric field.     

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.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

High-speed self-similar debonding of an interface between two orthotropic materials: The anti-plane shear case

The anti-plane problem of the transient debonding of an interface between two orthotropic materials is examined. The material principal axes are allowed to be oblique to the interface. The debonding is modeled as an interface crack propagating self-similarly from zero-length. The extending speed is assumed to be subsonic, transonic or supersonic. We first consider the dynamic debonding under the moving concentrated loading. The moving dislocation model of self-similar propagation of an interface crack is used to formulate the problem in a singular integral equation which is solved analytically. The stress singularity at the crack tips is discussed. The order of singularity is found to be one-half for subsonic debonding and to vary between zero and one-half depending on the crack speeds for transonic debonding. The dynamic stress intensity factors/coefficients for these two situations are presented in closed-form. The paper also concludes that supersonic debonding is impossible unless the loads are directly applied to the crack tips. Finally, the results for dynamic debonding under xⁿ-type loads are presented by using the superposition method. This revised version was published online in August 2006 with corrections to the Cover Date.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Stress intensity factor for a crack normal to an interface between two orthotropic materials

In this paper, the problem of a crack normal to an interface in two joined orthotropic plates is studied as a plane problem. Body force method is used to investigate dependence of the stress intensity factor on the elastic constants: E _x1, E _y1, G _xy1, V _xy1 for material 1 and E _x2, E _y2, G _xy2, V _xy2 for material 2. A particular attention is paid to simplifying kernel functions, which is used in the body force method, so that all the elastic constants involved can be represented by three new parameters: H ₁, H _2I, H ₃ for the mode I deformation and H ₁, H _2II, H ₃ for the mode II deformation. From the kernel function so obtained it is found that the effects of the eight elastic constants on the stress intensity factors can be expressed by the three material parameters, H ₁, H _2I, H ₃ and H ₁, H _2II, H ₃, respectively for K _I and K _II. Furthermore, it is also found that the dependence of K _I on H ₁, H _2I, H ₃ is exactly the same as the dependence of K _II on H ₁, H _2II, H ₃. This revised version was published online in August 2006 with corrections to the Cover Date.