Dissimilar Piezoelectric Materials Research Articles

Interfacial dynamic impermeable crack analysis in dissimilar piezoelectric materials by a new interaction integral

Dynamic intensity factors (IFs) provide an important parameter for assessing the failure risk of an interfacial crack for piezoelectric composites exposed to electromechanical impact loadings. In the present research, a new dynamic interaction integral (I-integral) is built for computing the dynamic stress IFs (SIFs) and electric displacement IF (EDIF) of an interfacial crack located in dissimilar inhomogeneous piezoelectric media. Through proper selection of auxiliary variables, the domain expression of I-integral for the inhomogeneous piezoelectric bi-materials does not need to take into account any derivative term for any piezoelectric material property. Further, after rigorous theoretical derivation, the resulting I-integral is still valid for complex models where the integration domain contains other multiple interfaces, regardless of whether the extra materials are straight or curved. Incorporating the modified extended finite element method, the accuracy is confirmed by checking the dynamic IFs extracted from the I-integral with referenced results. The domain-independence is tested by varying ranges of integration domains for inhomogeneous piezoelectric bi-materials and multi-interface piezoelectric composites. Finally, typical examples are employed to discuss the influences of the direction and magnitude of electric impact loading, combination of polarizations, inhomogeneous degree of piezoelectric bi-materials and complex distribution of diverse material properties on the dynamic IFs.

A set of electrically conducting collinear cracks between two dissimilar piezoelectric materials

The plane problem of multiple collinear cracks between two piezoelectric semi-infinite spaces is considered. The cracks can have arbitrary lengths and distances between each other and they are assumed to be electrically conductive. The materials are polarized in the orthogonal to the interface direction. In-plane combined tension-shear loading and the electric field parallel to the crack faces are prescribed at infinity. The Riemann-Hilbert problems of linear relationship are formulated and solved analytically by using the presentations of electro-mechanical quantities via sectionally-analytic functions. The arbitrary constants are found from the conditions at infinity and the system of linear algebraic equations which dimension is equal to the number of cracks. Stresses, electric field outside cracks as well as mechanical and electrical displacement jumps along the crack regions are found in the relatively simple analytical forms. Special attention was devoted to the energy release rate (ERR) determination. Due to analysis of asymptotic fields at the tips of single and multipleinterface cracks, the analytical presentations of the ERR for any crack tips are obtained. Numerical illustrations for several interface cracks are presented. The variations of stresses, electric field and the crack opening along the interface are demonstrated. The ERR are given for different crack tips, loading, crack locations and different physical aspects of the interaction effect are discussed. The particular cases of purely elastic and purely electrically conductive states are considered and good agreement with known results is demonstrated.

Periodic interfacial cracks in dissimilar piezoelectric materials under the influence of Maxwell stress

This paper presents an analytical method to investigate the dissimilar piezoelectric materials containing periodic cracks in the presence of uniform electromechanical loads with consideration of the Maxwell stress. Using Stroh’s formalism and conformal mapping, electric and stress fields are obtained for the semi-permeable cracks with a constant dielectric parameter. An approximate method is developed to obtain the analytical solution to the crack opening displacement, and which is validated by the pre-existing solution and with numerical integral. Using the method developed in this paper, the results of the electric displacement field in cracks, the stress, and electric displacement intensity factors are obtained. The numerical results show uniform electric displacement field inside in cracks, and it is independent of the distribution or the size of cracks when under static uniform electromechanical loads. Additionally, it is found that the field intensity is roughly the same as that in the case of a single interfacial crack when the gap of periodic crack exceeds four times the crack length.

Dynamic Response of Three-Dimensional Multi-Domain Piezoelectric Structures via BEM

In this paper, a Laplace domain boundary element method is applied for transient dynamic analysis of three-dimensional multi-domain linear piezoelectric structures. Piezoelectric materials of homogeneous sub-domains may have arbitrary degree of anisotropy. The boundary element formulation is based on a weakly singular representation of the piezoelectric boundary integral equations in the Laplace domain. To compute the time-domain solutions a convolution quadrature formula is applied for the numerical inversion of Laplace transform. Presented multi-domain boundary element method is tested on a three-dimensional problem of nonhomogeneous column which is made of two dissimilar piezoelectric materials and subjected to dynamic impact loading.

Multiple moving interfacial cracks between two dissimilar piezoelectric layers under electromechanical loading

The dynamic problem of several moving cracks at the interface between two dissimilar piezoelectric materials is analyzed. The combined out-of-plane mechanical and in-plane electrical loads are applied to the layers. Fourier transforms are used to reduce the problem to a system of singular integral equations with simple Cauchy kernel. The integral equations are solved numerically by converting to a system of linear algebraic equations and by using a collocation technique. The results presented consist of the stress intensity factors and the electric displacement intensity factors. It is found that generally the field intensity factors increase with increasing crack propagation speed.

Analysis of symmetric and skew-symmetric weight functions for a semi-infinite interfacial crack in transversely isotropic piezoelectric bimaterials

This paper presents an analysis of symmetric and skew-symmetric weight functions for in-plane interfacial crack problems in piezoelectric bimaterials. The symmetric weight function matrix is obtained by means of the solution of a Wiener–Hopf functional equation and the skew-symmetric weight function matrix is developed by the construction of a corresponding full field solution. The explicit expressions for the symmetric and skew-symmetric weight functions are given and applied to in-plane deformation problems for a semi-infinite crack between two dissimilar piezoelectric materials. The validity of the present method in the determination of the field intensity factors is demonstrated by illustrative examples, in which both piezoelectric bimaterials and piezoelectric/elastic bimaterials are considered. It is shown that, for the present interfacial crack problem, both symmetric and skew-symmetric weight function matrices are necessary in the general integral formula for the evaluation of field intensity factors, and that the contribution of the skew-symmetric element of the applied load is not negligible in fracture analysis.

Influences of the remanent polarization and Maxwell stress in surrounding medium on a moving anti-plane crack between two dissimilar piezoelectric solids

In this work, it is focused on the problems that the influences of the remanent polarization and Maxwell stress in the surrounding medium on a moving anti-plane interfacial crack between two dissimilar piezoelectric materials. By the Fourier and its inverse transforms, it is converted to two pairs of dual integral equations. The closed-form solutions of the intensity factors of the stress and electric displacement are derived. It can be observed that the difference between the moving crack and static case can be reduced by the remanent polarization. Influences of the Maxwell stress become more obvious for the systems with larger remanent polarizations. The fracture properties of the moving interfacial crack between two dissimilar piezoelectric solids can be obviously changed by the remanent polarization and Maxwell stress.

Interfacial dynamic impermeable cracks analysis in dissimilar piezoelectric materials under coupled electromechanical loading with the extended finite element method

Interfacial dynamic impermeable cracks analysis of dissimilar piezoelectric solids under coupled electromechanical impact loadings by the extended finite element method (X-FEM) is presented. The dynamic X-FEM approach recently developed by the authors is further extended to analyze transient responses of interfacial impermeable cracks in dissimilar piezoelectric structures. The mechanical displacements and electrical potential are approximated by appropriate enrichment functions that are not only for the crack-face and crack-tips, but also for the materials interfaces. In this work, we develop a new set of electro-mechanical enrichment functions for interfacial crack-tip in piezoelectric bimaterials, which is modified from the pure mechanical twelve-fold interfacial crack-tip enrichment functions used for mechanical problems. Unlike the pure mechanical set, the new modified enrichment functions contain the electric-mechanical factor, which is determined from the electro-mechanical fields of interfacial cracks in piezoelectric bimaterials. We also present asymptotic crack-tip fields in piezoelectric bimaterials based on the Stroh’s formalism, and then apply them in the evaluation of generalized dynamic intensity factors (GDIFs) via the interaction integrals. The accuracy of the proposed approach is validated by comparing the obtained GDIFs with the ones derived from boundary element method. Numerical dynamic results of interfacial cracks in piezoelectric bimaterials are investigated and some aspects of the influences of poling angles, interface inclination, impact loading, etc. on the GDIFs are analyzed. There are many important issues have found such as the accuracy of the GDIFs obtained by the present formulation is high; a pure electrical impact immediately induces non-zero GDIFs, while the elastic waves caused by the mechanical impact need some time to reach and excite the crack; the effect of polarization on the GDIFs is significant; the behaviors of the GDIFs are complicated and their peak values decrease with increasing poling angles; the interface inclination angle of cracks greatly alters the GDIFs; and the interfacial crack growth may be delayed either by the effects of polarization or the electrical impacts.

The extended finite element method with new crack-tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

Summary This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X-FEM) with new crack-tip enrichment functions. In the X-FEM, crack modeling is facilitated by adding a discontinuous function and crack-tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ϵ class and κ class, two classes of crack-tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J-integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

Moving Dugdale type crack along the interface of two dissimilar piezoelectric materials

In this paper we extend the concept of Dugdale crack model and Yoffe model to propose a moving Dugdale interfacial crack model, and the interfacial crack between dissimilar piezoelectric materials under anti-plane electro-mechanical loading is investigated considering the electro-mechanical nonlinearity. It is assumed that the constant moving crack is electrically permeable and the length of the crack keeps constant. Fourier transform is applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. The explicit expression of the yield zone size is derived and the crack sliding displacement has been explicitly obtained. The results show that both the stress and electric field in the cracked piezoelectric material are of finite value and the crack sliding displacement is dependent on the loading, material properties and crack moving velocity. The static interfacial crack problem can be recovered when the moving velocity is zero.

Analysis of three collinear antiplane interfacial cracks in dissimilar piezoelectric materials under non-self equilibrated electromechanical loadings on a center crack

The problem of three collinear interfacial cracks between two dissimilar transversely isotropic piezoelectric materials is considered under electromechanical loadings. The crack surfaces are assumed to be impermeable to the electric field. A single antiplane mechanical and inplane electrical loads are applied at a point on centred crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. By solving the problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green’s function for different loading conditions.

Stress intensity factor analysis of an interfacial corner between piezoelectric bimaterials using the H-integral method

Asymptotic solutions around the interfacial corner between piezoelectric bimaterials can be obtained by the combination of the Stroh formalism and the Williams eigenfunction expansion method. Based on an extension of the Stroh formalism and the H-integral derived from Betti’s reciprocal principle for piezoelectric problems, we analyzed the stress intensity factors (SIFs) and asymptotic solutions of piezoelectric bimaterials. The eigenvalues and eigenvectors of an interfacial corner between dissimilar piezoelectric anisotropic materials are determined using the key matrix. The H-integral for piezoelectric problems is introduced to obtain the scalar coefficients, which are related to the SIFs. We propose a new definition of the SIFs of an interfacial corner for piezoelectric materials, and we demonstrated the accuracy of the SIFs by comparing the asymptotic solutions with the results obtained by the finite element method (FEM) with very fine meshes.

Dynamic Anti-Plane Characteristic of Two Dissimilar Piezoelectric Media with an Interfacial Crack

Dynamic anti-plane characteristic is investigated theoretically on two dissimilar piezoelectric media with an interfacial crack subjected to time-harmonic incident anti-plane shearing in this paper. The formulations are based on the method of complex variable and Green’s function. Dynamic stress intensity factors at the crack’s tip are obtained by solving boundary value problems with the methods of conjunction and crack-division technique. The calculating results are plotted to show how the frequencies of incident wave, all kinds physical parameters of two dissimilar piezoelectric materials, applied electric loads and the dimension of the interfacial crack influence upon the dynamic stress intensity factor (DSIF). And some of the calculating results are compared with other published documents.

Electroelastic gap waves between dissimilar piezoelectric materials in different classes of symmetry

Shear surface waves guided by a gap between dissimilar piezoelectric half-spaces that belong to different classes of material symmetry are systematically investigated. It is shown that at most two surface waves can be guided by the structure. Equations for calculating surface wave velocities of propagation and corresponding existence conditions are obtained in closed form. For some special cases, surface wave velocities are derived explicitly. It is shown that despite the presence of a characteristic length in the problem (that is, the gap width) a non-dispersive surface wave independent of the wavelength and gap width can be guided by the structure. Electroelastic wave components and existence conditions of this non-dispersive wave are obtained. It is demonstrated that both the gap width and the dielectric permittivity of the gap have significant influences on the dynamic behaviour of the structure so that even small gaps cannot always be simplified and modelled as permeable, impermeable or absorbent with infinitesimal width.

The role played by the Coulombic traction for an interface crack in dissimilar piezoelectricmaterials

The role played by the Coulombic traction for an interface crack in dissimilar piezoelectricmaterials is clarified. Based on the extended Stroh theory, the Coulombic traction, usuallyneglected in piezoelectric fracture, is imposed on the interface crack surfaces. It is foundthat the low-capacitance medium (air or vacuum) inside the crack gap yields somelarge Coulombic traction as compared to the applied mechanical loading whetherthe remanent polarization of piezoelectric material is considered or not. Thus,previous investigations based on the traction-free condition underestimate therole of the Coulombic traction and in turn may yield unexpected errors for theeffective stress intensity factor (SIF) and energy release rate (ERR) at the cracktip.

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.

A Crack in an Electrode Layer between Two Dissimilar Piezoelectric Materials

A crack on the conductive interface between two dissimilar piezoelectric ceramics under electromechanical loading is investigated. The closed form of the singular crack tip fields for the interface crack is derived here using an analysis based on analytic functions. It is shown that the interfacial crack-tip field consists of a pair of oscillatory singularities. A closed form of the solution for a finite crack on the conductive interface between dissimilar piezoelectric media is also derived.

Analysis of a Thin Electrode Layer between Two Dissimilar Piezoelectric Solids

A thin electrode layer at the interface between two dissimilar linear piezoelectric materials under electromechanical loading is investigated. The complex function theory is employed to obtain the exact solution to a finite thin conductive layer. Special consideration is devoted to the structure of singular stress and electric fields near the tip of the thin electrode between two dissimilar piezoelectric materials. The stress and electric field are found to have an inverse square root singularity. The electric field intensity factor characterizes uniquely the singular fields close to the edge of the conductive line sheet.

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.

Solution for a Semi-Permeable Interface Crack Between Two Dissimilar Piezoelectric Material

A semi-permeable interface crack in dissimilar piezoelectric materials is studied in detail. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and the crack tip energy release rate (ERR). The Stroh complex variable theory (Stroh, A. N., 1958, Philos. Mag. 3, pp. 625–646;Ting, T. C. T., Int. J. Solids Struct., 22, pp. 965–983) is used to obtain the solution, from which some useful numerical results for 21 kinds of dissimilar piezoelectric materials are calculated. They are combined from seven kinds of commercial piezoelectric ceramics. The distribution of the normal electric displacement component (NEDC) along the interface crack is assumed to be uniform and the corresponding problem is then deduced to a Hilbert problem with an unknown NEDC. Solving the Hilbert problem and determining the near-tip field for each of the 21 bimaterials, we determine the crack tip singularities and find that the crack-tip singularity for a certain combination of two dissimilar piezoelectric materials can be either oscillatory or nonoscillatory when the poling axes of both piezoelectric materials are perpendicular to the interface crack. Energy analyses for PZT‐4∕BaTiO3 as a typical nonoscillatory class bimaterial and those for PZT-5H∕BaTiO3 as a typical oscillatory class bimaterial are specially studied in detail under four different conditions: (i) the crack gap is filled with air or vacuum; (ii) the crack gap is filled with silicon oil to avoid discharge; (iii) the crack gap is conducting; and (iv) the electrically impermeable crack. Detailed comparisons are performed among the four cases. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR under the combined electromechanical loading. We also conclude that the previous investigations under the insulating crack model are incorrect or misleading since the model overestimates the effect of the electric field on the ERR very much and the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack. Whereas the previous investigations under the conducting crack model may be accepted in a tolerant, way, the results of the ERR show very small discrepancies from those for the semi-permeable crack model, especially when it filled with silicon oil.