Effects of electric field and poling on the fatigue of cracked piezoelectric ceramics in cyclic three-point bending

Abstract

This paper deals with the fatigue behavior of cracked piezoelectric ceramics in cyclic bending under electric fields both numerically and experimentally. Fatigue tests were carried out in three-point bending with the single-edge precracked-beam specimens. The crack was created normal to the poling direction. Number of cycles to failure was measured under different electric fields. A plane strain finite element analysis was also performed, and the effect of polarization switching on the energy release rate was discussed under a high negative electric field. In addition, possible mechanisms for crack growth were discussed by scanning electron microscope examination of the fracture surface of the piezoelectric ceramics.

Appendix

For piezoelectric ceramics which exhibit symmetry of a hexagonal crystal of class 6 mm with respect to principal $x_{1}, x_{2}$ and $x_{3}$ axes, the constitutive relations can be written in the following form:

$$\begin{array}{rll}\left\{ \begin{array}{l} \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6} \end{array}\right\} &=& \left[ \begin{array}{cccccc} s_{11} & s_{12} & s_{13} & 0 & 0 & 0 \\ s_{12} & s_{11} & s_{13} & 0 & 0 & 0 \\ s_{13} & s_{13} & s_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & s_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & s_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & s_{66} \end{array}\right] \left\{ \begin{array}{l} \sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6} \end{array}\right\} \\ && + \left[ \begin{array}{lll} 0 & 0 & d_{31}\\ 0 & 0 & d_{31}\\ 0 & 0 & d_{33}\\ 0 & d_{15} & 0\\ d_{15} & 0 & 0\\ 0 & 0 & 0 \end{array}\right] \left\{ \begin{array}{l} E_{1}\\ E_{2}\\ E_{3} \end{array}\right\} + \left\{ \begin{array}{l} \varepsilon_{1}^{r} \\ \varepsilon_{2}^{r} \\ \varepsilon_{3}^{r} \\ \varepsilon_{4}^{r} \\ \varepsilon_{5}^{r} \\ \varepsilon_{6}^{r} \end{array}\right\}\end{array} $$

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$$\begin{array}{rll}\left\{ \begin{array}{l} D_{1}\\D_{2}\\D_{3} \end{array}\right\} &=& \left[ \begin{array}{cccccc} 0&0&0&0&d_{15}&0\\ 0&0&0&d_{15}&0&0\\ d_{31}&d_{31}&d_{33}&0&0&0 \end{array}\right] \left\{ \begin{array}{c} \sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6} \end{array}\right\} \\ && + \left[ \begin{array}{ccc} \epsilon_{11}^{T}&0&0\\ 0&\epsilon_{11}^{T}&0\\ 0&0&\epsilon_{33}^{T} \end{array}\right] \left\{ \begin{array}{c} E_{1}\\ E_{2}\\ E_{3} \end{array}\right\} + \left\{ \begin{array}{c} P_{1}^{r} \\ P_{2}^{r} \\ P_{3}^{r} \end{array}\right\}\end{array} $$

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where

$$\left. \begin{array}{l} \sigma_{1}=\sigma_{11},\ \sigma_{2}=\sigma_{22}, \ \sigma_{3}=\sigma_{33} \\ \sigma_{4}=\sigma_{23}=\sigma_{32}, \ \sigma_{5}=\sigma_{31}=\sigma_{13}, \ \sigma_{6}=\sigma_{12}=\sigma_{21} \end{array} \right\}$$

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$$\left. \begin{array}{l} \varepsilon_{1}=\varepsilon_{11},\ \varepsilon_{2}=\varepsilon_{22}, \ \varepsilon_{3}=\varepsilon_{33} \\ \varepsilon_{4}=2\varepsilon_{23}=2\varepsilon_{32}, \ \varepsilon_{5}=2\varepsilon_{31}=2\varepsilon_{13}, \ \varepsilon_{6}=2\varepsilon_{12}=2\varepsilon_{21} \end{array} \right\}$$

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$$\left. \begin{array}{l} \varepsilon_{1}^{r}=\varepsilon_{11}^{r},\ \varepsilon_{2}^{r}=\varepsilon_{22}^{r}, \ \varepsilon_{3}^{r}=\varepsilon_{33}^{r} \\ \varepsilon_{4}^{r}=2\varepsilon_{23}^{r}=2\varepsilon_{32}^{r}, \ \varepsilon_{5}^{r}=2\varepsilon_{31}^{r}=2\varepsilon_{13}^{r}, \ \varepsilon_{6}^{r}=2\varepsilon_{12}^{r}=2\varepsilon_{21}^{r} \end{array} \right\}$$

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$$\left. \begin{array}{l} s_{11}=s_{1111}=s_{2222},\ s_{12}=s_{1122}, \ s_{13}=s_{1133}=s_{2233},\ \\ s_{33}=s_{3333},\ s_{44}=4s_{2323}=4s_{3131},\ \\ s_{66}=4s_{1212} =2(s_{11}-s_{12}) \end{array} \right\}$$

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$$ d_{15}=2d_{131}=2d_{223}, \ \ d_{31}=d_{311}=d_{322}, \ \ d_{33}=d_{333} $$

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