Mathematical Problems in Engineering

Volume 2015, Article ID 371083, 10 pages

http://dx.doi.org/10.1155/2015/371083

## Cell-Based Smoothed Finite Element Method-Virtual Crack Closure Technique for a Piezoelectric Material of Crack

School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China

Received 17 December 2014; Revised 5 February 2015; Accepted 5 February 2015

Academic Editor: Timon Rabczuk

Copyright © 2015 Li Ming Zhou 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

In order to improve the accuracy and efficiency of solving fracture parameters of piezoelectric materials, a piezoelectric element, tailored for the virtual crack closure technique (VCCT), was used to study piezoelectric materials containing a crack. Recently, the cell-based smoothed finite element method (CSFEM) and VCCT have been used to simulate the fracture mechanics of piezoelectric materials. A center cracked piezoelectric materials with different material properties, crack length, mesh, and smoothing subcells at various strain energy release rates are discussed and compared with finite element method-virtual crack closure technique (FEM-VCCT). Numerical examples show that CSFEM-VCCT gives an improved simulation compared to FEM-VCCT, which generally simulates materials as too stiff with lower accuracy and efficiency. Due to its simplicity, the VCCT piezoelectric element demonstrated in this study could be a potential tool for engineers to practice piezoelectric fracture analysis. CSFEM-VCCT is an efficient numerical method for fracture analysis of piezoelectric materials.

#### 1. Introduction

Piezoelectric materials have been widely used in high technology fields due to their attractive electromechanical coupling characteristics. Piezoelectric materials are typically brittle materials. Therefore, pores and cracks often arise in their manufacture or application process due to the electromechanical joint effect. The main cause of cracks is material failure. Solving the fracture parameters of piezoelectric materials accurately will have significant impact on their applications and may lead to device performance improvements.

Pak [1], Sosa [2], Suo et al. [3], Wang [4], and Zhang and Hack [5] began research on the fracture mechanics of piezoelectric materials in the early 1990s and have since become the focus of attention in this field [6–9]. In the 20 years to date, wider research has been conducted by both domestic and foreign researchers, with a remarkable progress as a result. The theoretical framework of the fracture mechanics of piezoelectric materials has been established. However, the theoretical model applies only to simple questions and in order to solve more complex problems, one still has to resort to numerical methods. The first significant numerical attempt using finite element implementation for piezoelectric phenomenon was a piezoelectric vibration analysis proposed by Allik and Hughes [10].

Until now, displacement finite element method (FEM) models have been used mostly for engineering problems. However, it is well known that FEM produces overestimations of the stiffness matrix [11, 12]. As a consequence, the solution is always smaller than the real result. Additionally, since mapping and coordinate transforms are involved in the FEM, elements are not allowed to be of arbitrary shape. In the effort of overcoming these problems, Liu et al. proposed for the first time a cell-based smoothed finite element method (CSFEM) by combining the existing FEM technology with the strain smoothing technique of mesh-free methods [13]. No derivative of the shape functions is involved in computing the field gradients to form the stiffness matrix. Correspondingly, the element shape in CSFEM can be of arbitrary shape. In CSFEM, the strain in an element is modified by smoothing the compatible strains over quadrilateral smoothing domains, which gives important softening effects. CSFEM can improve the accuracy and convergence rate of the FEM-Q4 model using the same mesh. The SFEM was extended to various problems such as shells [14], piezoelectric material [15], fracture mechanics [16], heat transfer [17], and structural acoustics [18] among others. CS-FEM has been combined with the extended FEM to address problems involving discontinuities.

Rabczuk et al. [19] presented an extension of the phantom-node method by allowing crack tips to be placed within a finite element. Thereby, the crack growth in the phantom-node method became almost independent of the finite element mesh. Wu et al. [20] applied the NMM to investigate the cracking behavior of a sedimentary rock under dynamic loading. By incorporating the NMM with the cracking processes, crack initiation, propagation, and coalescence were successfully modeled. The element free Galerkin method (EFGM) [21, 22], developed by Belytschko et al., has a unique feature in solving the problems of crack growth. The notable feature of this method is that there is no mesh required in establishing a discrete equation. Moreover, it only needs to arrange discrete points in the global domain. Thus, the complicated process of mesh formation is avoided and influences from mesh distortion are reduced. A new method for treating crack growth by particle methods has been proposed by Rabczuk and Belytschko [23]. The crack is treated as a collection of cracked particles. At each cracked particle, a discontinuity along a line in 2D or a plane in 3D is introduced, where the normal depends on the complete constitutive model of the material. Shi et al. [24] presented an extended meshless method based on the partition of units used for concurrent multiple crack situations and multiple crack simulations. This method describes the discontinuous displacement field and crack tip singularity field caused by embedding discontinuous items and the crack tip singularity field function into a conventional meshless approximation function. Nanthakumar et al. [25] developed an algorithm to detect and quantify defects in piezoelectric plates. The inverse problem is solved iteratively where XFEM is used for solving the forward problem in each iteration. Béchet et al. [26] applied XFEM to the fracture of piezoelectric materials. Nguyen-Vinh et al. [27] present an extended finite element formulation for dynamic fracture of piezoelectric materials.

VCCT was put forward in 1977 by Rybicki and Kanninen [28]. Xie and Biggers [29, 30] had done a lot of research work for VCCT. Compared with the extrapolation method and local or entire equivalent domain integrals, VCCT has an obvious advantage in solving fracture parameters [31–33]. It only uses the nodal force and displacement to calculate the strain energy release rate and only requires a single step in the numerical analysis, thereby simplifying the problem and giving the additional advantages such as high precision and efficiency, no need for special processing of the crack tip unit and small grid size requirements [29, 34, 35]. To date, there are no reports on the virtual crack closure of the electromechanical coupling field.

In this paper, a piezoelectric element tailored for VCCT was used to study the crack of piezoelectric materials. CSFEM and VCCT were introduced into fracture mechanics of piezoelectric materials and CSFEM-VCCT for piezoelectric material with cracks was put forward. The energy release rates of different piezoelectric materials with cracks are discussed and compared with FEM-VCCT.

#### 2. Governing Equations

The constitutive equations for a two-dimensional piezoelectric material in the - axis can be expressed in terms of the strains and the electric field:where , , , and are the stress tensor, the strain tensor, the electric displacement vector, and the electric field vector, respectively. , , and are the elastic stiffness, piezoelectric, and dielectric constants, respectively.

The strain matrix is related to displacements by

The strain displacement relation can be expressed using the condensed matrix notation given in [13]where and are the displacement in the - and -directions, respectively. Commas followed by indices represent differentiation with respect to that index (i.e., ).

The electric field is related to electric potential byThe mechanical equilibrium is governed byAnd the governing electrostatic equilibrium is given by

The two-dimensional matrix form of the mechanical and electrical constitutive equations is given by [15]where are the elastic compliance constants, are piezoelectric constants, and are the dielectric constants. The superscript represents quantities measured at constant stress.

The finite element solution for 2D piezoelectric problems using the standard linear element can be expressed aswhere is the number of nodes of an element; , are shape function matrices; and and are the nodal displacement and nodal electric potential vectors, respectively.

The corresponding approximations of the linear strain and electric field arewhere

Using Hamilton’s principle, the piezoelectric static equations of an element can be obtained as follows:in which

#### 3. Cell-Based Smoothed Finite Element Method

In the stabilized conforming nodal integration technique, the strain and the electric field used to evaluate the stiffness matrix are computed by a weighted average of the standard strain and electric field of the finite element method. In particular, at an arbitrary point in a smoothing element domain , they are approximated as follows:where is a constant smoothing function described bywhere is the area of the smoothing cell . The cell-based element approach is illustrated in detail in Figure 1.