Mathematical Problems in Engineering

Volume 2015, Article ID 638783, 13 pages

http://dx.doi.org/10.1155/2015/638783

## Enriched Element-Free Galerkin Method for Fracture Analysis of Functionally Graded Piezoelectric Materials

School of Mechanical Science and Engineering, Jilin University, No. 5988 Renmin Street, Changchun 130022, China

Received 16 January 2015; Revised 30 March 2015; Accepted 15 April 2015

Academic Editor: Timon Rabczuk

Copyright © 2015 Guang Wei Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new method using the enriched element-free Galerkin method (EEFGM) to model functionally graded piezoelectric materials (FGPMs) with cracks was presented. To improve the solution accuracy, extended terms were introduced into the approximation function of the conventional element-free Galerkin method (EFGM) to describe the displacement and electric fields near the crack. Compared with the conventional EFGM, the new approach requires smaller domain to describe the crack-tip singular field. Additionally, the domain of the nodes was not affected by the crack. Therefore, the visibility method and the diffraction method were no longer needed. The mechanical response of FGPM was discussed, when its material parameters changed exponentially in a certain direction. The modified -integrals for FGPM were deduced, whose results were compared with the results of the conventional EFGM and the analytical solution. Numerical example results illustrated that this method is feasible and precise.

#### 1. Introduction

Functionally graded materials (FGMs) are composite materials formed of two or more constituent phases with a continuously variable composition. During design, the requirements of structural strength, reliability, and lifetime of piezoelectric structures/components call for enhanced mechanical performance, including stress and deformation distribution under multifield loading. In recent years, the emergence of FGMs has demonstrated that they have the potential to reduce stress concentration and to provide improved residual stress distribution, enhanced thermal properties, and higher fracture toughness. Consequently, a new kind of material, functionally graded piezoelectric materials (FGPMs), has been developed to improve the reliability of piezoelectric structures by introducing the concept of the well-known FGM to piezoelectric materials [1].

At present, FGPMs are usually associated with particulate composites where the volume fraction of particles varies in one or several directions. One of the advantages of a monotonous variation of volume fraction of constituent phases is elimination of the stress discontinuities that are often encountered in laminated composites and accordingly avoidance of delamination-related problems. How all these aspects can be improved and what the mechanisms might be are popular topics which have received much attention from researchers. Wang and Noda [2] investigated the thermally induced fracture of a functionally graded piezoelectric layer bonded to a metal. Ueda studied the fracture of an FGPM strip with a normal crack [3, 4], a symmetrical FGPM strip with a center crack [5], and an FGPM strip with a two-dimensional crack [6, 7]. Li and Weng [8] solved the problem of an FGPM strip containing a finite crack normal to boundary surfaces. Hu et al. [9] studied the problem of a crack located in a functionally graded piezoelectric interlayer between two dissimilar homogenous piezoelectric half planes. Rao and Kuna [10] presented an interaction integral method for computing stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) in FGPM under thermoelectromechanical loading. Borrelli et al. [11] used the energy-decay inequality technique to analyze the decay behavior of end effects in antiplane shear deformation in piezoelectric solids and FGPMs. Zhong and Shang [12] developed an exact solution for a functionally graded piezothermoelectric rectangular plate. Dai et al. [13] conducted a theoretical study of electromagnetoelastic behavior for an FGPM cylinder and sphere. They then extended their solutions to include thermal effects [14]. Zhong and Yu [15] analyzed the FGPM beam with arbitrarily graded material properties along the beam thickness direction. Based on the layerwise finite element model, Shakeri and Mirzaeifar [16] performed a static and dynamic analysis of a thick FGM plate with piezoelectric layers. Wang et al. [17] analytically investigated the axisymmetric bending of circular plates whose material properties varied with the thickness. Using the Fourier transform technique, Chue and Yeh [18] developed a system of singular integral equations for angle cracks in two bonded FGPMs under antiplane shear. Chen and Bian [19] studied the wave propagation characteristics of an axially polarized, functionally graded, piezoceramic cylindrical transducer submerged in an infinite fluid medium. Ueda [20] addressed the problem of two coplanar cracks in an FGPM strip under transient thermal loading. Ben Salah et al. [21] examined the propagation of ultrasonic guided waves in FGPMs. Wang et al. [22] studied the singularity behavior of electroelastic fields in a wedge with angularly graded piezoelectric material (AGPM) under antiplane deformation. Chue and Yeh [23] extended the results of the case of two arbitrarily oriented cracks in two bonded FGM strips.

In the field of engineering technology, accurate solutions computed with analytic methods are only available in problems with relatively simple equations and regular geometry. As for most problems, especially in the case of complex engineering, analytic solutions cannot be computed. Therefore, research has been going on for many years that has led to the development of another approach—the numerical method. With the rapid development and wide application of computers, numerical analysis and theoretical investigation and experimental investigation are considered as the three major research approaches. As one of the most effective tools for the study of mechanics, the finite element and other numerical methods have been widely used in scientific research and engineering practice. FEM analysis of piezoelectric structures with a crack under dynamic electromechanical loading was presented by Enderlein et al. [24]. A survey on numerical algorithms for crack analyses in piezoelectric structures to be used with FEM for determining fracture parameters was presented by Kuna [25]. Béchet et al. [26] presented an application of XFEM to the analysis of fracture in piezoelectric materials. Nanthakumar et al. [27] analyzed the multiple flaws in piezoelectric structures using XFEM and level sets. Sharma et al. [28] analyzed a subinterface crack in piezoelectric bimaterials with XFEM. Bouvier et al. [29] studied the inverse problems in structural analysis: application to atherosclerotic plaque elasticity reconstruction by using the XFEM.

Compared with the extended finite element method, however, element-free method has a unique feature in solving the problems of crack growth, plastic flow of materials, geometric distortion and phase transition, and singularity. The notable feature of this method is that, in establishing the discrete equation, it does not need mesh but only needs to arrange discrete points in the global domain. Thus, it not only avoids the complicated process of mesh formation, but also greatly reduces the influence of mesh distortion. Various meshless methods have been applied to the analysis of smart materials and structures such as the meshless point collocation method (PCM) [30], the point interpolation meshfree method (PIM) [31], a novel truly hybrid meshless-differential order reduction method (hM-DOR) [32], and the local Petrov-Galerkin method (MLPG) [33]. Among these meshless methods, the element-free Galerkin method (EFGM) [34], developed by Belytschko et al., has good compatibility and stability and will not have the problem of shear locking of volume even in adopting linear primary function. And fast convergence speed and high accuracy can be achieved. EFGM was widely applied in fracture mechanics. Rabczuk and Belytschko analyzed the problem of a three-dimensional large deformation meshfree method for arbitrary evolving cracks [35]. Rabczuk et al. deduced a simplified meshfree method for shear bands with cohesive surfaces [36].

In this study, a type of electromechanical-coupling enriched element-free Galerkin method based on the enriched EFG methods [37, 38] is developed. Enriched terms were introduced into the approximation function of the conventional EFGM to describe the displacement and electric fields near the crack. Compared with the conventional EFGM, this method only needs a small domain to describe the crack-tip singular field. Furthermore, the domain of the node is not affected by the crack without using the visibility method and diffraction methods.

#### 2. Basic Equations for Two-Dimensional FGPM

The governing equations and the boundary conditions of FGPM are briefly given.

*Constitutive Equations*. Considerin which , , and are the elastic, piezoelectric, and dielectric constants, respectively. , , , and are the stress tensor, strain tensor, electric field, and electrical displacements.

Strains are related to displacements by the expressionwhere and are, respectively, the displacements in - and -directions.

The electric field is related to the electric potential by the expression

*Types of Gradation*. The properties of FGM are usually assumed to have the same functions of certain space coordinates. Exponential material gradation is commonly used graded forms.

All the material constants including elastic constants, piezoelectric parameters, and dielectric constants follow the exponential law:where represents material constants such as , , or , is the corresponding value at the plane , and *α* denotes a material graded parameter.

The field equations of electroelasticity are reduced to two-dimensional form in the special cases: plane strain. Considering a transversely isotropic FGPM, according to (1), the - plane is the isotropic plane, and one can employ either the - or - plane for the study of plane electromechanical phenomena. The plane strain conditions require thatBy substitution of (10) into (1), we haveor inverselywhere , , and are the reduced material constants. They are related to the elastic compliance tensor , the piezoelectric tensor , and the dielectric impermeability tensor by the following relations [39]:

*Boundary Conditions*. In electroelasticity theory, mechanical boundary conditions are formulated just as in classical elasticity theory. The electric boundary conditions are, however, still controversial. The first attempt to define the electric boundary conditions over crack faces was done by Parton [40]. He assumed that although the magnitude of the normal electrical displacement component at the crack face was very small, the electrical displacement was continuous across the crack faces. He used the following electric boundary conditions:Later, Hao and Shen [41] improved the above assumption by taking the electric permeability of air in the crack gap into consideration. In addition to (14), they presented an equation for the boundary condition at crack faces:where is the permittivity of air. However, (15) has remained disregarded for a long time due to its complex mathematical treatment.

As pointed out by Suo et al. [42], the above assumption is not physically practical as there will clearly be a potential drop across the lower capacitance crack. This is particularly true for those piezoelectric ceramics with permittivity 10^{3} times higher than that in the air. For this reason, Deeg [43] proposed another set of electric boundary conditions over crack faces:Equation (16) is derived from the constitutive equation .

This is equivalent to having the crack surfaces free of surface charge which is the electrical boundary condition. Thus the electric displacement vanishes in the environment.

#### 3. Electromechanical-Coupling Enriched Element-Free Galerkin Method

Displacement and electric potential were adopted as basic field quantities for the solution of the enriched element-free Galerkin method, and the displacement and electric potential interpolation of a typical point were conducted as follows:where is the MLS shape function at the point . is the support domain. is a set of all nodes whose support is cut by the crack. The set is a set of all nodes that lie within a fixed region around the crack tip. and are the additional degrees of freedom of the displacement. and are the additional degrees of freedom of the electric potential.

The first terms of (15) and (16) are the conventional EFGM approximation which simulates the displacement and electric potential fields. The second term is the displacement and electric potential approximation function of the nodes in . is the Heaviside function:where is the projection of point on the crack.

The last term in (15) and (16) is the displacement and electric potential approximation function of the nodes in . is the branch functions given bywhere and are polar coordinates in the local coordinate system.

The set Ω_{Γ} includes the nodes whose support contains point or is cut by the crack. The set is nodes whose support contains point and crack tip (see Figure 1).