Fatigue crack analysis in piezoelectric specimens by a single-domain BEM
Introduction
Because of the extensive applications of piezoelectric intelligent devices and the frequent failure behaviors in actuators, the piezoelectric fracture problems have caused the attentions of many scientists in the past few decades [47]. Plenteous analytic solutions have been obtained for classic crack problems in piezoelectrics [11], [14], [40], [45]. For more detailed reviews who can refer to Kuna [26] and Fang and Liu [12]. Due to the limitation of the current analytical methods, more general cases should turn to numerical methods. The classical finite element method (FEM) has been frequently used for crack analysis in piezoelectric structures [24], [25]. By introducing of enrichment functions to model discontinuities, the extended finite element method (XFEM) was successfully applied to analyze the static and dynamic fracture problems in piezoelectrics [2], [5], [50]. The recently rapid-developed meshless methods have also been extended to this problem. For example, Sladek et al. [44] proposed a meshless local Petrov-Galerkin method (MLPG) for crack analysis in functionally graded piezoelectric materials.
Although various new numerical methods have been developed for modeling crack problems, the classical boundary element method (BEM) is still the one of the most efficient numerical methods for crack analysis. This is because of its major characteristics of semi-analytic and dimensionality reduction. Compared to the other numerical approaches, more stable and precise results could be obtained by the BEM. Only crack boundary is need to be discretized, which make this method very suitable for simulating crack propagation. Accordingly, it has got a number of successful applications in piezoelectric fracture mechanics. Lee and Jiang [27] first derived the fundamental solution for plane piezoelectricity by using the double Fourier transform and set up the boundary integral equations (BIEs) by the method of weighted residuals. Pan [34] first applied a single-domain BEM to fracture analysis in anisotropic piezoelectric solids and derived the Green's functions for the anisotropic infinite, semi-infinite and bi-material piezoelectric plane by using the complex variable function method. Sáez and co-workers [17], [46] extended the single-domain BEM for dynamic fracture problems in piezoelectrics. Lei and co-workers [28], [31] derived an explicit formula for calculating the field intensity factors of interfacial piezoelectric cracks and carried out some successful crack analyses by the BEM.
Most existing studies focused on the stationary crack analysis in piezoelectric solids, only a few attempts have been made to simulate the crack evolution by numerical methods. The FEM is the mostly exploited method [22], [23]. Jański and Kuna [19] simulated the propagating cracks in piezoelectric specimens with an adaptive FEM program. Bui [4] studied the quasi-static crack propagation in piezoelectric solids using the extended isogeometric analysis (XIGA). Recently, a single-domain BEM was applied to crack propagation analysis in piezoelectric plates by Lei et al. [29]. Although Bhattacharya et al. [3] presented a paper with a title of dealing with the fatigue crack growth in piezoelectric material by the XFEM, the fact is, only static crack growth was considered in that paper. To the authors’ knowledge, few numerical work on fatigue cracks in piezoelectric materials has been reported till now.
The fracture criteria play important roles on modeling the evolution of cracks. Several criteria have been proposed for piezoelectric materials, e.g. the total energy release rate (ERR, G), the mechanical ERR (GM) [37], the local ERR GC [15], the generalized stress intensity factor (SIF) and the crack opening displacement (COD) [11], energy density criterion [43], the modified hoop SIF Kωω [48], etc. As stated by Fang et al. [11], the calculated critical fracture loads based on the COD, GM and GC criteria are well agreed with their experimental data. Although the maximum-Kωω criterion was adopted in some numerical methods to determine the crack growth direction, Lei and Zhang [32] observed that a positive electric field will decrease the maximum of Kωω, which contradicts some existing experiments. Although the GM-based criterion has been verified by Fang et al. [11], it is time-consuming to find its maximum by using its original definition along the crack plane, which retards its numerical application. To overcome this drawback, Lei and Zhang [32] recently proposed a hoop MERR which is just related to the current crack-tip. It has been verified to be a good approximation of the MERR for a sufficiently small kink at any angle to the current crack tip, which is of high efficiency for numerical analysis.
Piezoelectric ceramics tend to fatigue under cyclic electric field or mechanical loadings. Fatigue crack growth will occur at lower SIFs than crack growth observed under static loading [42]. Some experiment tests have been done for the mechanic or electric induced fatigue crack problems [6], [13], [33], [42]. But few numerical studies have been developed. The objective of this work is to develop an effective single-domain BEM for modeling fatigue crack growth in piezoelectric structures. Here, we further applied the -based facture criterion to determine the direction of the crack growth. The constants in the Paris’ laws of different fracture parameters were obtained by fitting the existing experimental data. By using them, the remaining lives of some pre-cracked piezoelectric specimens were analyzed by the BEM. All the involved fracture parameters were computed by the interaction integral method. Some numerical examples for fatigue crack growth analysis under cyclic mechanical and electrical loadings were respectively considered. The results were discussed in detail and some interesting conclusions were presented finally.
Section snippets
A single-domain BEM for crack problems in piezoelectric materials
When a closed crack model is considered by the classical BEM, the mathematical degeneration will arise from the two coplanar crack surfaces [9]. Some special techniques should be adopted to overcome this difficulty. One direct way is by dividing the problem domain into two subdomains along the crack faces, called sub-region method, see Aliabadi and Rooke [1]. Then, the classical displacement boundary integral equation (BIE) is applied to each subdomain boundary, respectively. But it encounters
Computation of the fracture parameters
In this paper, all the fracture parameters involving in the fracture criteria and the Paris-type laws are computed by the interaction integral as described in Lei et al. [30]. The interaction integral is derived from the J-integral by superposing two admissible states, i.e., actual and auxiliary states. Let the two states (1) and (2) be denoted by the superscripts of the field variables. The J-integral for the superimposed state (s) can be expressed aswhere
Determination of crack growth direction
As mentioned above, there exist several criteria for determination of the crack growth direction in piezoelectric solids under coupled electromechanical loadings. Although the mechanical energy release rate is one of the effective fracture parameters which has been verified by some experiments (see [11]), it's time-consuming for finding its extreme value, which retards its numerical application. To overcome this, the following apparent hoop MERR was proposed by Lei and Zhang [32]
Results and discussion
The fatigue crack growth induced by cyclic mechanical loading or electric field in PZT specimens will be analyzed by the single-domain BEM program in the following numerical calculations. All examples are assumed to be under the plane-stress conditions. The material constants are listed in Table 1 for these PZT ceramics used in this paper. To avoid the possible numerical truncation errors caused by the huge differences among the magnitudes of these material constants in piezoelectricity, a
Conclusions
In this paper, we have developed a single-domain boundary element computer program and successfully applied it for fatigue crack analysis in piezoelectric specimens under cyclic mechanical loading or electric field. The direction of the crack extension was determined by the maximum hoop ERR criterion. The crack growth length was determined by some Paris-type laws. Through the numerical experiments, some major conclusions can be drawn as follows:
- (1)
For a cracked piezoelectric specimen under
Acknowledgments
This work is supported by the Natural Science Foundation of China under Grant No. 11472021, and the German Research Foundation (DFG) under the project number ZH 15/14-1, which are gratefully acknowledged.
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