A generalized self-consistent method for piezoelectric fiber reinforced composites under antiplane shear

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Abstract

A three-phase piezoelectric confocal elliptical cylinder model is proposed, and an exact solution is obtained for the model subjected to antiplane mechanical and inplane electrical loads at infinity by using the conformal mapping integrated with the Laurent series expansion technique. Based on the model and solution, a generalized self-consistent method is developed for predicting the relevant effective electroelastic moduli of piezoelectric fiber reinforced composite, accounting for variations in fiber section shapes and randomness in distribution and orientation. The dilute, self-consistent, differential and Mori–Tanaka micromechanics theories for piezoelectric fiber reinforced composites are also extended to consider randomness in fiber section orientation in a statistical sense. A full comparison is made among these five micromechanics methods and with the Hori and Nemat-Nasser's rigorous upper and lower bounds, which shows that the generalized self-consistent method and Mori–Tanaka method can verify each other's results, whereas other micromechanics methods may lead to significant deviations, or even unacceptable results. Finally, as an application of the proposed generalized self-consistent method, the complex factors that influence the effective piezoelectric modulus are discussed.

Introduction

Applications of piezoelectric solids have increased dramatically in recent years, fueled largely by their wide uses in smart composite materials and structures. Their intrinsic electro-mechanical coupling behavior presents a level of difficulty not present in design and analysis of mechanical behavior of composite materials.

Many theoretical studies have focused on the micromechanics models of heterogeneous piezoelectricity and the predictions of effective electroelastic moduli for piezoelectric composites. The references listed herein (Grekov et al., 1989, Olson and Avellaneda, 1992, Dunn and Taya, 1993, Benveniste, 1994a, Benveniste, 1994b, Chen, 1994, Li and Dunn, 1999) are some examples of the contributions in this area. The dilute, self-consistent, differential and Mori–Tanaka methods are the micromechanics ones that have received the most attention and use. They are all based on the two-phase model of inclusion/matrix. By using the four micromechanics methods, Dunn and Taya (1993) examine carefully the predictions of the effective electroelastic properties of piezoelectric composites. They pointed out that the four methods give close predictions at lower volume fractions which are about less 0.2, but the predictions significantly diverge at higher volume fractions. In light of the existing experimental data, it cannot be ascertained which prediction will be better. The fact retards the routine applications of the micromechanics methods in engineering to some extent.

The generalized self-consistent method is a more sophisticated micromechanics approach (Christensen and Lo, 1979, Christensen and Lo, 1986, Luo and Weng, 1987, Christensen, 1993, Huang and Hu, 1995, Jiang and Cheung, 1998, Riccardi and Montheilet, 1999, Jiang and Cheung, 2001). Different from the aforementioned micromechanics methods based on the two-phase model, the generalized self-consistent method is based on the three-phase model, i.e., an inclusion and a surrounding ring matrix constitute a representative unit cell, which, in turn, is embedded in an infinite composite. The existing generalized self-consistent method was developed for non-piezoelectric composites. It is reported (for example, refer to Huang et al., 1994) that the difference between the generalized self-consistent method and Mori–Tanaka method is small for predicting uncoupled effective mechanics properties, and that the former is in even better agreement with experimental data. The three-phase model is also used to improve the accuracy of the Mori–Tanaka method (Luo and Weng, 1987). Obviously, it is highly desirable to develop a generalized self-consistent method based on the three-phase model for piezoelectric composites so that a full comparison to the aforementioned micromechanics methods based on a two-phase model can be made. This is the objective of the present work.

The early generalized self-consistent method can only accommodate circular and spherical inclusions, Huang and Hu (1995), Jiang and Cheung (1998), and Riccardi and Montheilet (1999) extended it to cover elliptical and spheroiodal inclusions. In this paper, a three-phase piezoelectric confocal elliptical cylinder model with any section orientation is proposed. An exact solution is obtained for the model subjected to antiplane mechanical and inplane electrical loads at infintity by using the conformal mapping integrated with the Laurent expansion technique. Based on the model and solution, a generalized self-consistent method is developed for predicting the relevant electroelastic moduli of piezoelectric fiber reinforced composites, accounting for variations in fiber section shapes and randomness in distribution and orientation.

The existing dilute, self-consistent, differential and Mori–Tanaka methods for piezoelectric composites (Dunn and Taya, 1993) accommodate monotonically aligned piezoelectric inclusions. In this paper, the formulation of Dunn and Taya (1993) is further extended to cover the case of transversely randomly oriented piezoelectric elliptical fibers. It is noted that Zhao and Weng (1990) extended the Mori–Tanaka method to estimate the effective elastic behavior for conventional (non-piezoelectric) composites with two-dimensional randomly oriented elliptic fibers.

Another important approach for predicting effective properties of heterogeneous materials is to estimate upper and lower bounds for the effective properties, which is pioneered by Hashin and Shtrikman (1962). Later many scholars make contributions to this approach. For examples, see recent works by Nemat-Nasser and Hori (1995), Balendran and Nemat-Nasser (1995) and Munashinghe et al. (1996). In previous researches, essentially the same procedure has been applied to predict bounds for both (uncoupled) mechanical and non-mechanical properties. However, much less attention has been paid to the coupled mechanical and non-mechanical properties. Hori and Nemat-Nasser (1998) consider the case of piezoelectricity as an illustrative example of predicting bounds for coupled mechanical and non-mechanical properties. Rigorous upper and lower bounds for the effective moduli are obtained for heterogeneous piezoelectric materials. In this paper, Hori and Nemat-Nasser's bounds are used to examine the accuracy of the above-mentioned five micromechanics methods.

The factors that influence the effective piezoelectric modulus are of practical importance in design, manufacture and use of piezoelectric composites. As an application of the present method, this problem is discussed.

Section snippets

Model and basic formulation

Fig. 1 is a schematic diagram of the three-phase confocal elliptical piezoelectric cylinder model. The elliptical region Sf encircled by L1 represents the piezoelectric fiber cross-section and the elliptical ring region Sm between L1 and L2 represents the piezoelectric matrix in the representative unit cell. L1 and L2 share the common foci O1 and O2. According to the generalized self-consistent method, the volume fraction of the piezoelectric fiber in the representative unit cell is equal to

Complex potential solution

Eq. (10) shows that the general solution of the generalized displacement vector U can be expressed by an analytical function vector or a complex potential vector Φ*(z), where z=x1+ix2 is the complex variable in the local coordinates (Fig. 1).U=ReΦ*(z),where Re denotes the real part, andΦ*(z)=Φ*w(z)Φ*ϕ(z),Φ*w(z) and Φ*ϕ(z) are conventional analytical functions. By using the complex potential vector, the constitutive Eq. (8) can be expressed asΣ1=MReΦ*′(z),Σ2=−MImΦ*′(z)orΣ1iΣ2=M(Z1iZ2)=*′(z),

Generalized self-consistent method

In this section, the matrix equation will be derived for predicting the effective electroelastic moduli by the generalized self-consistent method. From Eq. (18), it is seen that the remaining work is to determine the averaged generalized stresses.

Substituting , , , into , , then into Eq. (22), we obtain the generalized stresses in the fiber and matrix of the representative unit cell in the local coordinate system:Σ1iΣ2f=Mf(Q1cosβ+iQ2sinβ)Σ0,Σ1iΣ2m=MmQ3−(Q52)cosβ+iQ4−(Q62)sinβ1−(1/ζ2)Σ0.

Other micromechanics methods

The micromechanics methods that have received the most attention and used are the dilute, self-consistent, Mori–Tanaka and differential methods. These methods have been applied to estimate the coupled electroelastic behavior of piezoelectric composites with monotonically aligned reinforcements by Dunn and Taya (1993). Zhao and Weng (1990) extended the Mori–Tanaka method for conventional (non-piezoelectric) composites with monotonically aligned elliptical fibers to consider the case of

Numerical results and comparison

In this section, a full numerical comparison of the developed generalized self-consistent method to other four micromechanics methods will be made, and Hori and Nemat-Nasser's (1998) bounds are also used to examine the accuracy of the five micromechanics methods.

Consider PZT-7A fiber/epoxy composite with constituent material properties:C44f=25.4GPa,e15f=9.2C/m2,d11f=4.071nC2/Nm2,C44m=1.8GPa,e15m=0,d11m=0.03717nC2/Nm2,where the superscripts f and m refer to the piezoelectric fiber and matrix,

Conclusions

A three-phase piezoelectric confocal elliptical cylinder model is proposed. An exact solution is obtained for the model subjected to antiplane mechanical and inplane electrical loads at infinity. Based on the model and solution, a generalized self-consistent method is developed for predicting relevant electroelastic moduli of piezoelectric fiber reinforced composites.

The dilute, self-consistent, differential and Mori–Tanaka methods for piezoelectric composites with monotonically aligned

Acknowledgements

The work is supported by the Hong Kong Research Grants Council, the National Natural Science Foundation of China and the Aviation Science Foundation of China.

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