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TECHNICAL PAPERS

# Solution for a Semi-Permeable Interface Crack Between Two Dissimilar Piezoelectric Material

[+] Author and Article Information
Q. Li

School of Aerospace,  Xian Jiao-Tong University, Xian City 710049, P.R. China

Y. H. Chen1

School of Aerospace,  Xian Jiao-Tong University, Xian City 710049, P.R. Chinayhchen2@mail.xjtu.edu.cn

1

Corresponding author.

J. Appl. Mech 74(5), 833-844 (Sep 27, 2006) (12 pages) doi:10.1115/1.2711232 History: Received April 28, 2006; Revised September 27, 2006

## Abstract

A semi-permeable interface crack in dissimilar piezoelectric materials is studied in detail. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and the crack tip energy release rate (ERR). The Stroh complex variable theory (Stroh, A. N., 1958, Philos. Mag.3, pp. 625–646;Ting, T. C. T., Int. J. Solids Struct., 22, pp. 965–983) is used to obtain the solution, from which some useful numerical results for 21 kinds of dissimilar piezoelectric materials are calculated. They are combined from seven kinds of commercial piezoelectric ceramics. The distribution of the normal electric displacement component (NEDC) along the interface crack is assumed to be uniform and the corresponding problem is then deduced to a Hilbert problem with an unknown NEDC. Solving the Hilbert problem and determining the near-tip field for each of the 21 bimaterials, we determine the crack tip singularities and find that the crack-tip singularity for a certain combination of two dissimilar piezoelectric materials can be either oscillatory or nonoscillatory when the poling axes of both piezoelectric materials are perpendicular to the interface crack. Energy analyses for $PZT‐4∕BaTiO3$ as a typical nonoscillatory class bimaterial and those for $PZT–5H∕BaTiO3$ as a typical oscillatory class bimaterial are specially studied in detail under four different conditions: (i) the crack gap is filled with air or vacuum; (ii) the crack gap is filled with silicon oil to avoid discharge; (iii) the crack gap is conducting; and (iv) the electrically impermeable crack. Detailed comparisons are performed among the four cases. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR under the combined electromechanical loading. We also conclude that the previous investigations under the insulating crack model are incorrect or misleading since the model overestimates the effect of the electric field on the ERR very much and the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack. Whereas the previous investigations under the conducting crack model may be accepted in a tolerant, way, the results of the ERR show very small discrepancies from those for the semi-permeable crack model, especially when it filled with silicon oil.

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## Figures

Figure 1

A semi-permeable interface crack of length 2a between two dissimilar piezoelectric materials subjected to the far field mechanical and electric loading

Figure 2

The crack tip energy release rate for an interface crack in PZT–4∕BaTiO3 bimaterial subjected to the remote tensile stress and electric field. Plots (a), (b), (c), and (d) are given for four different levels of applied tensile stress: 0MPa, 2MPa, 5MPa, and 10MPa, respectively. Here, curves with the symbol ∎ refer to the impermeable crack, curves with the symbol ● refer to the semi-permeable crack filled with air or vacuum, curves with the symbol ▾ refer to the semi-permeable crack filled with silicon oil, and curves with the symbol ▴ refer to the permeable crack.

Figure 3

The crack tip energy release rate for an interface crack in PZT-5H∕BaTiO3 bimaterial subjected to the remote tensile stress and electric field. Plots (a), (b), (c), and (d) are given for four different levels of applied tensile stress 0MPa, 2MPa, 5MPa, and 10MPa, respectively. Here, curves with the symbol ∎ refer to the impermeable crack, curves with the symbol ● refer to the semi-permeable crack filled with air or vacuum, curves with the symbol ▾ refer to the semi-permeable crack filled with silicon oil, and curves with the symbol ▴ refer to the permeable crack.

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