Cracks In Piezoelectric Materials Research Articles

Effect of quadratically varying electric potential on the multiple cracks in piezoelectric materials

A basic solution to the problem of three collinear equal cracks in a transversely isotropic piezoelectric material plate is obtained under the influence of electromechanical forces. Stroh formalism and complex variable methods are used to derive the mathematical expressions for SIF (stress intensity factor), COD (crack opening displacement), and COP (crack opening potential). The length of the saturation zones is investigated at each crack tip under the influence of quadratically varying electric displacement. Furthermore, a numerical study is carried out to discuss the effect of quadratically varying electrical potential on the saturation zone, COD, and COP. The results obtained numerically are reported graphically.

Strip electro-mechanical yield model for three equal straight collinear cracks in piezoelectric materials

This paper is devoted to the development of a model to study three straight collinear cracks in piezoelectric materials under the influence of combined electrical and mechanical loading conditions. Mathematical relationships for the size of yield zones with mechanical yielding conditions and the saturation zone sizes to applied electrical field has been established. To make this plausible, two different cases are considered: first when the electrical saturation zones are bigger than the mechanical yield zones while other case in which situation is just opposite. Closed-form analytic expressions are derived for various fracture parameters with the help of well known complex variable method. Numerical results are presented to study the multiple cracks in different piezoelectric materials. Results obtained are reported graphically, analyzed and concluded.

XFEM for semipermeable crack in piezoelectric material with Maxwell stress

In this work, extended finite element method (XFEM) is implemented to model the semipermeable cracks in the piezoelectric material with Maxwell stresses. This Maxwell stress is induced due to the presence of an electric field at the crack faces and boundary of the piezoelectric material. To model the semipermeable crack with Maxwell stress, the crack tip is enriched with six-fold enrichment functions. The electro-mechanical interaction integral is modified and implemented for several crack configurations to compute the stress intensity factor (SIF). The accuracy of the XFEM-based semipermeable crack model in the presence of Maxwell stresses is demonstrated by validating with the analytical solution. The effects of crack length, far-field loading, Maxwell stress, and poling direction are investigated on the normalized SIF.

A XFEM approach for the three-dimensional cracks in piezoelectric material using interaction integral

Piezoelectric materials are widely used in various engineering applications such as electromechanical actuators, vibrators, energy harvesters, sensors, transducers, and propeller blades. These materials are subjected to various types of cyclic loading and their crack growth analysis is quite useful as it predicts the material’s life. This work proposes an extended finite element method to examine the static three-dimensional cracks in the piezoelectric material. The modeling of a crack is done using crack front elements enriched with the new six folded enrichment functions. These functions have been derived from the analytical solution of semi-infinite crack in a piezoelectric domain and expanded using Laurent-like series. Domain-based interaction integral is obtained to extract the mixed-mode stress intensity factors. Further, the auxiliary mechanical and the electrical fields in the interaction integral are obtained based on Stroh formulation. The XFEM is employed to evaluate SIF and EDIF in a three-dimensional domain is implemented in MATLAB code. The XFEM formulation is validated with static planar penny shaped crack for Mode I SIF and Mode IV EDIF with the standard solutions available in the literature. The error in Mode I and Mode IV is 0.0476 and 0.1444 respectively. The XFEM prediction values matches well with the analytical predictions. Further, simulation of the other cracks like inclined penny shaped crack, lens-shaped crack, and elliptical crack have been carried out with an expectation of usefulness of present methodology to mechanical engineers and designers.

Dynamic intensity factors for multiple permeable cracks in piezoelectric materials

AbstractThis article presents the dynamic intensity factors for multiple permeable Mode‐I cracks in piezoelectric materials plane by means of the generalized Almansi's theorem. The Fourier transform is used and the present problem is formulated into two pairs of dual integral equations. The dual integral equations are solved by the Schmidt method. The dynamic stress intensity factors (DSIFs) and the dynamic electric displacement intensity factor (DEDIF) at the crack tips are given. It is shown that the effects of the crack length, the distance between two parallel cracks and the circular frequency of incident wave near the crack tips on the DSIFs and DEDIF.

Numerical Simulation of Tri-layer Interface Cracks in Piezoelectric Materials Using Extended Finite Element Method

Piezoelectric materials have wide applications due to their special electromechanical coupling characteristics but are vulnerable to sudden failure because of their brittle nature. Researchers have been profoundly interested in investigating the fracture of piezoelectric materials under different applied conditions. In the present work, tri-layer plates with interface cracks in piezoelectric materials have been analyzed using XFEM. Impermeable static cracks at the interfaces of PZT-5H and PZT-4 have been investigated under mechanical and electromechanical loading conditions. Numerical results show that the ɛ (oscillating) singularity vanishes and $$\kappa$$ class singularity exists for the current combination of materials. Enrichment functions for $$\kappa$$ class are applied to obtain the stress and electric fields near the crack tip. An interaction integral approach is used to evaluate fracture parameters. An effect on stress and electric displacement intensity factors due to variation in mechanical and electric loading has also been investigated.

Dynamic analytical solution of a limited-permeable mode-I crack in piezoelectric materials based on the non-local theory

This paper presents the dynamic non-local theory solution to a limited-permeable mode-I crack in piezoelectric materials (PMs) with the crack surfaces subjected to harmonic stress waves. The generalized Almansi’s theorem and the Schmidt method are used for this purpose. Based on the Fourier transform technique, this problem is formulated into coupled dual integral equations. The dynamic non-local stress and the dynamic non-local electric displacement fields at the crack tips are obtained by solving the derived dual integral equations. Numerical results are reported and discussed to illustrate the influences of the crack length, the lattice parameter, the characteristics of the harmonic wave and the electric permittivity of the air inside the crack on the dynamic non-local stress field and the dynamic non-local electric displacement field near the crack tips in PMs. As compared to the known electrically permeable crack mode, the proposed one exhibits more applicability.

A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method

Based on the concept of the hoop field intensity factors of an initial crack prior to any kink, an apparent hoop mechanical (strain) energy release rate (MERR) is defined to approximate the MERR of a piezoelectric crack with an infinitesimal kink at any arbitrary angle. The validity and the efficiency of the simplified approximation are examined by numerical examples using the boundary element method (BEM). The generalized crack-opening-displacements or displacement jumps are computed by the traction boundary integral equations (BIEs). By using the displacement extrapolation method, the crack-tip field intensity factors of any arbitrarily kinked crack in linear piezoelectric materials are obtained and the BEM results are validated by comparing them with the available reference analytical results. Then, the differences between the conventional field intensity factors and MERR of an infinitesimally kinked crack and the hoop field intensity factors and hoop MERR of the main crack prior to any kink are numerically analyzed. Finally, the crack propagation in an infinite linear piezoelectric material is numerically simulated. The paths of the crack growth are predicted by adopting four different fracture criteria, namely, the maximum hoop stress intensity factor (SIF) and MERR fracture criteria for the main crack-tip before the next propagation, and the maximum KI and MERR fracture criteria for the kinked tip of the main crack with an infinitesimal branch at an arbitrary kinking angle evaluated by using a trial crack extension technique. The comparisons among these results show that the present simplified approximation can efficiently provide a sufficient accuracy for numerical simulation of crack growth in linear piezoelectric materials.

Multiple cracks identification for piezoelectric structures

We propose a novel iterative procedure to solve the inverse problem of detecting multiple cracks in two-dimensional piezoelectric structures under dynamic electric loads. The effect of dynamic electric loads and a time-harmonic excitation with a single frequency is investigated. The crack is detected through the solution of an inverse time-dependent problem. The extended finite element method (XFEM) is employed for solving the forward problem as it allows the use of a single regular mesh for a large number of iterations with different crack geometries. The Newmark-beta method is employed for the time integration. In each iteration the forward problem is solved for various crack geometries, and at each iteration, the mechanical and electrical response of the piezoelectric structure is minimized at known specific points along the boundary to match the measured data. The minimization of the objective function is performed by multilevel coordinate search (MCS) method. The algorithm is an intermediate between purely heuristic methods and methods that allow an assessment of the quality of the minimum obtained and is in spirit similar to the direct method for global optimization. In this paper, the XFEM-MCS methodology under the dynamic electric load is applied to two-dimensional electromechanical problems where different number of straight cracks are considered. The results show that this methodology can be effectively employed for detecting multiple cracks in piezoelectric materials.

Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS

Abstract Accurate numerical modeling of multifield piezoelectric materials is challenging because of the inherent electro-mechanical coupling effect and material anisotropic behaviors. The modeling becomes even more difficult especially for problems with non-smooth solutions like crack under dynamic loading. We present in this paper an extension of the extended isogeometric analysis (XIGA) for simulation of two-dimensional fracture mechanics problems in piezoelectric materials under dynamic and static coupled electromechanical loads. The discretization of problem domain is based on basis functions generated from NURBS, which are used for both geometric description and approximation of solution field variables. To capture the discontinuity across the crack-faces and the singularity at the crack-tip, the isogeometric approximation is locally enriched by discontinuous Heaviside function and asymptotic crack-tip branch functions. The sixfold enrichment functions particularly suitable for electromechanical crack-tip singularity of piezoelectric materials are used. To evaluate the generalized fracture parameters, a domain-form of electromechanical interaction integral is employed. For dynamic analysis, the implicit time integration scheme considering inertial effect is taken. Five numerical examples for single and mixed-modes of impermeable cracks are considered and the generalized fracture parameters under dynamic and static loads are analyzed. The accuracy and effectiveness of the proposed XIGA are illustrated through numerical investigations of the generalized dynamic and static fracture parameters. Numerical results are validated against the reference solutions derived from the boundary element methods. The effects of some numerical aspect ratios on generalized fracture parameters are also investigated. Additionally, we present some numerical results of quasi-static crack propagation in piezoelectric solids using the developed XIGA, taking fracture toughness anisotropy of polarized electroelastic materials into account, and employing the maximum modified hoop stress intensity factor criterion for predicting the growing direction of crack.

Modeling of pre-fracture zones for limited permeable crack in piezoelectric materials

A plane-strain problem for a limited permeable crack in an adhesive thin interlayer between two semi-infinite piezoelectric spaces is considered. The tensile mechanical stress and the electric displacement are applied at infinity. The interlayer is assumed to be softer than the connected materials; therefore, the zones of mechanical yielding and electric saturations can arise at the crack tips on the continuations of the crack. These zones are considered in this work. It was assumed that the length of electric saturation zones is larger than the length of mechanical yield zones. The zones of mechanical yielding are modeled by the crack continuations with normal compressive stresses applied at its faces. The electric saturation zones are modeled by segments at the crack continuations with prescribed saturated electric displacements. These electric displacements can linearly vary along the mechanical yielding zones. The problem is reduced to the Hilbert–Riemann problem of linear relationship, which is solved exactly. The equation for the determination of the yielding zones length, the expressions for the crack-opening displacement jump, electric potential jump, and J-integral is obtained in an analytical form. In case of finite size body, the finite elements method is used and the variation in the fracture mechanical parameters with respect to this size is demonstrated.

Propagation of a semi-infinite conducting crack in piezoelectric materials: Mode-I problem

In this paper, the mode-I transient response of a semi-infinite conducting crack propagating in a piezoelectric material with hexagonal symmetry under normal impact loading is investigated. The integral transform methods together with the Wiener–Hopf technique are used to solve the mixed boundary value problem under consideration. The solutions of the coupled fields are derived for two cases, i.e., generalized Rayleigh wave exists or not. The dynamic stress intensity factor and dynamic electric displacement intensity factor as well as their universal functions are obtained in a closed form. The numerical results for two universal functions are provided to illustrate the characteristics of dynamic crack propagation. It is found that the universal functions for the dynamic stress and electric displacement intensity factors vanish when the crack propagation speed reaches the generalized Rayleigh speed which is the propagation speed of surface wave in a piezoelectric half-space with metallized surface. It is noted that the electromechanical coupling coefficient has an important influence on the dynamic fracture characteristics.

Contact Zone Approach for an Electrically Impermeable Crack in Piezoelectric Materials Under Thermal-Mechanical Loading

A partial contact zone model is developed for the stress and electric displacement fields due to the obstruction of a uniform heat flux by an electrically impermeable crack in piezoelectric materials. Green's function method is used to reduce the problem to a set of singular integral equations that are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one-half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is reformulated assuming a contact zone at one crack tip. The extent of this zone, the stress and electric displacement intensity factors at each crack tip are obtained as functions of the applied mechanical stress and heat flux.

압전재료에서 비정상적으로 전파하는 투과형 모드 III 균열

면외 기계적하중과 면내 전기적하중을 받는 압전재료에서 비정상적으로 전파하는 전기투과형 균열에 대하여 연구하였다. 비정상적으로 전파하는 균열에 대한 평형방정식을 개발하였고 근접해법으로 투과형균열에 대한 응력장 및 변위장을 얻었다. 압전상수, 유전율 그리고 균열선단속도 및 응력확대계수의 시간 변화율이 비정상적으로 전파하는 균열선단부근의 응력 및 변위장에 미치는 영향을 명확히 나타내었다. 이러한 응력장과 변위장을 사용하여 비정상적으로 전파하는 균열선단의 응력장 및 변위장의 특성에 대하여 토론하였다.

Extended finite element simulation of stationary dynamic cracks in piezoelectric solids under impact loading

This work presents a transient dynamic analysis of stationary cracks in two-dimensional, homogeneous and linear piezoelectric solids subjected to coupled electromechanical impact loads using the extended finite element method (X-FEM). To serve this purpose, a dynamic X-FEM computer code using quadrilateral elements in conjunction with the level set method to accurately describe the crack geometry is developed. The sixfold basis enrichment functions particularly suitable for cracks in piezoelectric materials are adopted to fully capture the singular fields at the crack-tips in piezoelectricity. The governing equations are transformed into a weak-form and the time-dependent system of discrete equations is then obtained, which is solved by the unconditionally implicit time integration method at each time-step. To accurately assess the relevant dynamic mechanical stress and electric displacement intensity factors precisely and efficiently, domain-form of the contour integration integral taking the inertial effect into account in conjunction with the asymptotic near crack-tip fields of piezoelectric materials is presented. Four numerical examples for stationary cracks in homogeneous piezoelectric solids with impermeable crack-face boundary condition under impact loads are considered, respectively. Validation of the present method is made by comparing the present results with reference solutions available in the literature, and very good agreements are obtained. The effects of different poling directions and combined electromechanical impact loads are analyzed in details.

A supplementary study on fracture tests of piezoelectric material: cracks parallel to the poling direction

This study focuses on the behavior of a crack in piezoelectric material in which the crack is parallel to the poling direction. Tests were carried out on four-point bend specimens made of PZT-5H (Morgan Electro Ceramics, Wrexham, UK). Cracks were introduced parallel to the poling direction. With an electric field induced perpendicular to the cracks, the load was increased until failure occurred. Using the load at fracture, the level of the electric field and the critical crack length, finite element analyses were carried out to determine the intensity factors. These included K I , K IV and a small K II component. The latter occurred because of the small asymmetry of the crack. The material within the crack (air) was modeled to be a dielectric material. A fracture criterion is implemented in which the test results show small scatter about the failure curve. These tests were carried out in order to improve on the scatter obtained from a previous set of tests presented in Motola et al. (Int J Fract 159:167–190, 2009a). The scatter from those results will be explored.

Basic solution of four 3‐D rectangular limited‐permeable cracks in piezoelectric materials

AbstractThe solution of four 3‐D rectangular limited‐permeable cracks in piezoelectric materials were given by using the generalized Almansi's theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack, the shape of the rectangular crack and the distance among four rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.

Energy density and release rate study of an elliptic-arc interface crack in piezoelectric solid under anti-plane shear

The anti-plane problem of an elliptical inhomogeneity with an interfacial crack in piezoelectric materials is investigated. The system is subjected to arbitrary singularity loads (point charge and anti-plane concentrated force) and remote anti-plane mechanical and in-plane electrical loads. Using the complex variable method, the explicit series form solutions for the complex potentials in the matrix and the inclusion regions are derived. The electroelastic field intensity factors, the corresponding energy release rates and the generalized strain energy density at the cracks tips are then provided. The influence of the aspect ratio of the ellipse, the crack geometry and the electromechanical coupling coefficient on the energy release rate and the strain energy density is discussed and shown in graphs. The results indicate that the energy release rate increases with increment of the aspect ratio of the ellipse and the influence of electromechanical coupling coefficient on the energy release rate is significant. The strain energy density decreases with increment of the aspect radio of the ellipse and it is always positive for the cases discussed. The energy release rate, however, can be negative when both mechanical and fields are applied.

Basic solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials

The solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials were given by using the generalized Almansi’s theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack,the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.

Axisymmetric Problem of Energetically Consistent Interacting Annular and Penny-Shaped Cracks in Piezoelectric Materials

The axisymmetric problem of a concentric set of energetically consistent annular and penny-shaped cracks in an infinite piezoelectric body subjected to uniform far-field electromechanical loading is addressed. With the aid of a robust innovated technique, the pertinent four-part mixed boundary value problem (MBVP) is reduced to a decoupled Fredholm integral equation of the second kind. The results of two limiting cases of a single penny-shaped crack and a single annular crack are recovered. The contour plots of dimensionless intensity factors (IFs) at each crack front provide the stress and electric displacement intensity factors (SIFs and EDIFs, respectively) for all combination of crack sizes. The impermeable, permeable, and semipermeable models are also examined as limiting cases.