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Research Papers

# $M$-Integral for Calculating Intensity Factors of Cracked Piezoelectric Materials Using the Exact Boundary Conditions

[+] Author and Article Information
Yael Motola

The Dreszer Fracture Mechanics Laboratory, School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israelymotola@eng.tau.ac.il

Leslie Banks-Sills

The Dreszer Fracture Mechanics Laboratory, School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel

J. Appl. Mech 76(1), 011004 (Oct 31, 2008) (9 pages) doi:10.1115/1.2998485 History: Received August 21, 2007; Revised May 11, 2008; Published October 31, 2008

## Abstract

In this paper, the $M$-integral is extended for calculating intensity factors for cracked piezoelectric ceramics using the exact boundary conditions on the crack faces. The poling direction is taken at an angle to the crack faces within the plane. Since an analytical solution exists, the problem of a finite length crack in an infinite body subjected to crack face traction and electric flux density is examined. In this case, poling is taken parallel to the crack faces. Numerical difficulties resulting from multiplication of large and small numbers were treated by normalizing the variables. This problem was solved with the $M$-integral and displacement-potential extrapolation methods. With this example, the superiority of the conservative integral is observed. The values for the intensity factor obtained with the $M$-integral are found to be more accurate than those found by means of the extrapolation method. In addition, a crack parallel to the poling direction in a four-point bend specimen subjected to both an applied load and an electric field was analyzed and different electric permittivity values in the crack gap were assumed. It is seen that the electric permittivity greatly influences the stress intensity factor $KII$ and the electric flux density intensity factor $KIV$. The absolute value of these intensity factors increases with an increase in the value of the electric permittivity in the crack. The influence of the permittivity on $KI$ is rather small.

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## Figures

Figure 7

Four-point bend specimen

Figure 8

Mesh in the vicinity of the crack for both meshes of the four-point bend specimen shown in Fig. 7 with a/W=0.2

Figure 1

Crack tip and material coordinates

Figure 2

Integration paths for J-integral calculation

Figure 3

Mesh and integration paths about the crack tip

Figure 4

Griffith crack problem

Figure 5

Meshes for the plate in Fig. 4: (a) coarse and (b) fine meshes

Figure 6

Meshes in the neighborhood of the crack tip for the infinite plate: (a) coarse and (b) fine meshes

Figure 9

Normalized intensity factors as a function of normalized crack length a/W as obtained from the fine mesh for the specimen in Fig. 7: (a) K̃I, (b) K̃II, and (c) K̃IV

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