Piezoelectric Bimaterials Research Articles

Theoretical transient analysis and wave propagation of piezoelectric bi-materials

The transient response of piezoelectric bi-materials subjected to a dynamic anti-plane concentrated force or electric charge with perfectly bonded interface is examined in the present study. The problem is solved by using the Laplace transform method and the inverse Laplace transform is evaluated by means of Cagniard’s method. Exact transient full-field solutions of the contribution for each wave are expressed in explicit closed forms. The transient behavior of field quantities is examined in detail by numerical calculations. The existence condition of a propagating surface wave along the interface is discussed in detail. A surface wave can be guided by the interface of two semi-infinite materials in contact if one, at least, of these two materials is piezoelectric. The propagation velocity of the surface wave is explicitly expressed and is found to be less than the lower shear wave velocity of the two materials. The existence of the surface wave for piezoelectric–piezoelectric bi-materials is restricted to the situation that the shear waves of the two piezoelectric materials are very close. The possibility for the existence of the surface wave for piezoelectric–elastic bi-materials is much greater than that of the piezoelectric–piezoelectric bi-materials.

On some relations between different interface crack models

AbstractExact analytical solutions for an interface crack with an artificial contact zone in isotropic, anisotropic and piezoelectric materials have been analyzed and their comparison with the correspondent classical results has been performed. This analysis showed the way of a presentation of the main results for an artificial contact zone model in the manner very similar to the classical model, and due to this phenomenon the essential simplification of the investigation of the contact zone model has been attained. The application of the obtained results to interface cracks in anisotropic and piezoelectric bimaterials can be demonstrated. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

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

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

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.

Eshelby problem of polygonal inclusions in anisotropic piezoelectric bimaterials

This paper studies the two–dimensional Eshelby problem in anisotropic piezoelectric bimaterials. Assuming that the inclusion is an arbitrarily shaped polygon with uniform eigenstrain and eigenelectric fields, we derive the exact closed–form solution by integrating analytically the line–source Green functions in the corresponding bimaterials. The required line–source Green functions are obtained in terms of the Stroh formalism and include six different interface models. Since the induced elastic and piezoelectric fields due to the eigenstrain and eigenelectric fields are given in the exact closed form in terms of simple elementary functions, those due to multiple inclusions can be superposed together. Benchmark numerical examples are also presented for the induced elastic and piezoelectric fields within a square inclusion due to a uniform hydrostatic eigenstrain with the bimaterials being made of typical quartz and ceramic.

Screw dislocation in two dissimilar piezoelectric layers

AbstractThe elastic and electrical fields produced by a straight screw dislocation in two dissimilar piezoelectric layers are derived through the use of conformal mapping technique, where the solution for a semi‐infinite piezoelectric bimaterial is used as a base. The expression for the image force acting on the dislocation is given explicitly. The equilibrium position of the dislocation is further determined by letting the image force be equal to zero. In the case where one of the two layers is pure elastic, the numerical results for the image force are provided to illustrate the influence of piezoelectric effect on the behavior of the dislocation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary integral–differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media

Using the method of Ding et al. [Int. J. Solids Struct. 34 (1997) 3041], the fundamental solutions for three-dimensional two-phase transversely isotropic piezoelectric media are re-derived. Based on the fundamental solutions, the displacements and the electric potential at any point for an internal crack parallel to the interface in a two-phase transversely isotropic piezoelectric medium are expressed in terms of the displacement and electric potential discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the displacement and electric potential discontinuities are obtained for arbitrarily shaped planar interfacial cracks in three-dimensional two-phase transversely isotropic piezoelectric media. A boundary element formulation to solve the boundary integral–differential equations is presented. Numerical results of stress and electric displacement intensity factors and energy release rate of penny shaped and elliptical cracks are presented to illustrate the application, accuracy and efficiency of the proposed method in analyzing interfacial cracks in three-dimensional transversely isotropic piezoelectric bimaterials.

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

AbstractAn interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters.A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.

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 ε (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 BaTiO 3, 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 ε=0 or κ=0) are also obtained numerically, and the result show that there still exist four linear independent associate eigenvectors for each bimaterial.

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 ε or κ. It is shown in the present paper that the related eigenvalue problems for either ε=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.

Moving screw dislocations in piezoelectric bimaterials

AbstractAn explicit solution for a moving screw dislocation in a piezoelectric bimaterial is given by means of complex variables. The electroelastic fields and forces acting on the moving dislocation are obtained in closed form. As special cases, solutions for a moving screw dislocation in a piezoelectric half‐plane with a free or rigid surface are obtained explicitly.

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings II. Electrically impermeable interface cracks

This paper constitutes the second part of a study of interface cracks with contact zones in thermopiezoelectrical bimaterials, and it is concerned with the case of an electrically impermeable interface crack. The principal physical peculiarity of this case in comparison with an impermeable interface crack is connected with the dependencies of the contact zone length and the fracture mechanical parameters on the prescribed electrical flux, and in a mathematical sense the main peculiarity is concerned with the reduction of the problem in question to the joint solution of inhomogeneous combined Dirichlet–Riemann and Hilbert boundary value problems. The exact analytical solutions of the mentioned problems have been found for an arbitrary contact zone length, and the required thermal, mechanical and electrical characteristics at the interface as well as the associated fracture mechanical parameters at the corresponding crack tips are presented. The transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. Using the admissible directions of the heat and the electrical fluxes defined in this paper as well, the dependencies of the real contact zone length and the associated fracture and electrical intensity factors on the intensities of the thermal and electrical fluxes are presented in tables and associated diagrams.

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings: I. Electrically permeable interface cracks

An electrically permeable interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading and a temperature flux is considered. Assuming that all fields are independent on the coordinate x 2 co-directed with the crack front, the stresses, the electrical and the temperature fluxes as well as the derivatives of the jumps of the displacements, the electrical potential and the temperature at the interface are presented via a set of analytic functions in the ( x 1, x 3)-plane with a cut along the crack. Due to this representation firstly an auxiliary problem concerning the direction of the heat flux permitting a transition from a perfect thermal contact to a separation has been solved for a piezoelectric bimaterial. Besides, an inhomogeneous combined Dirichlet–Riemann boundary value problem has been formulated and solved exactly for the above mentioned interface crack. Stress and electrical displacements intensity factors are found in a clear analytical form which is especially easier for a small contact zone length. A simple equation and a closed form analytical formula for the determination of the real contact zone length have been derived and compared with the associated equation of the classical (oscillating) interface crack model defining the zone of crack faces interpenetration. For a numerical illustration of the obtained results a bimaterial cadmium selenium/glass has been used, and the influence of the heat flux upon the contact zone length and the associated stress intensity factor has been shown.

A Half-Infinite Coupled Crack on an Interface of Piezoelectric Bimaterials Without Oscillation

ABSTRACTBased on Stroh's formalism, analytic solutions are derived for a half-infinite coupled crack in piezoelectric bimaterials without oscillation by using analytical function technique. Four intensity factors related to the crack tip fields are obtained. It is found that these intensity factors are independent of the bimaterials constants when no oscillation occurs. Some numerical calculations about the stress tensor and the electric field ahead of the crack tip are conducted finally.

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J 1-integral and the path-dependent of J 2-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.

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.

Mindlin's problem for an anisotropic piezoelectric half–space with general boundary conditions

This paper considers Mindlin's problem in an anisotropic and piezoelectric halfspace with general boundary conditions, including 16 different sets of surface conditions. The Green's function due to...

Three-dimensional Green’s functions in anisotropic piezoelectric bimaterials

In this paper, a recently proposed method by E. Pan and F.G. Yuan (Int. J. Solids Struct., 2000) for the calculation of the elastic bimaterial Green’s functions is extended to the analysis of three-dimensional Green’s functions for anisotropic piezoelectric bimaterials. The method is based on the Stroh formalism and two-dimensional Fourier transforms in combination with Mindlin’s superposition method. We first derive Green’s functions in exact form in the Fourier transform domain. When inverting the Fourier transform, a polar coordinate transform is introduced so that the radial integral from 0 to +∞ can be carried out exactly. Therefore, the bimaterial Green’s functions in the physical domain are derived as a sum of a full-space Green’s function and a complementary part. While the full-space Green’s function is in an explicit form, as derived recently by E. Pan and F. Tonon (Int. J. Solids Struct., 37 (2000): 943–958), the complementary part is expressed in terms of simple regular line integrals over [0, 2π] that are suitable for standard numerical integration. Furthermore, the present bimaterial Green’s functions can be reduced to the special cases such as half-space, surface, interfacial, and full-space Green’s functions. Uncoupled solutions for the purely elastic and purely electric case can also be simply obtained by setting the piezoelectric coefficients equal to zero. Numerical examples for Green’s functions are given for both half-space and bimaterial cases with transversely isotropic and anisotropic material properties to verify the applicability of the technique. Certain interesting features associated with these Green’s functions are observed and discussed, as related to the selected material properties.

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

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 x2, 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.

Crack path selection in piezoelectric bimaterials

The problem of crack path selection in piezoelectric bimaterials is considered in this paper. Based on the Stroh formulation for anisotropic material and Green’s functions for piezoelectric bimaterials, the crack problem is expressed in terms of coupled singular integral equations at first, and then the equations are used to solve for stress and electric displacement fields numerically. A crack impinging an interface joining two dissimilar materials may arrest or may advance by either penetrating the interface or deflecting into it. 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.