13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Null-field integral approach for the piezoelectricity problems with arbitrary elliptical inhomogeneities Ying-Te Lee1, Jeng-Tzong Chen1*, Shyh-Rong Kuo1 1 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan * Corresponding author: jtchen@mail.ntou.edu.tw Abstract Based on the successful experience of solving anti-plane problems containing arbitrary elliptical inclusions, we extend to deal with the piezoelectricity problems containing arbitrary elliptical inhomogeneities. In order to fully capture the elliptical geometry, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion. Only boundary nodes are required instead of boundary elements. Therefore, the proposed approach belongs to one kind of meshless and semi-analytical methods. Besides, the error stems from the number of truncation terms of the eigenfuntion expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. It is worth noting that there are Jacobian terms in the degenerate kernel, boundary density and contour integral. However, they would cancel each other out in the process of the boundary contour integral. As the result, the orthogonal property of eigenfunction is preserved and the boundary integral can be easily calculated. Finally, the problem of two elliptical inhomogeneities in an infinite piezoelectric material subject to anti-plane remote shear and in-plane electric field is considered to demonstrate the validity of the present method. Besides, two circular inhomegenieties can be seen as a special case to compare with the available data by approximating the major and minor axes. Keywords Piezoelectricity, Elliptical inhomogeneity, In-plane electric field, Anti-plane shear 1. Introduction In recent years, more and more investigators paid their attention to study the actuators and sensors because they were widely used in smart materials or structures technology. Therefore, the study of electromechanical behavior of piezoelectric material becomes an important issue. It is well-known that it results in the stress concentration when the inhomogeneities or defects exist in the materials. In this article, we extend the previous works [1] on the piezoelectricity problems with “circular” inclusions to deal with the problem containing “elliptical” inhomogeneities. For an elliptical shape, it may be more general than a circular geometry in the practical applications. Based on the concept of complex potential, Gong and Meguid [2] used the conformal mapping and Laurent series expansion to solve an infinite medium containing an elliptical inhomogeneity under anti-plane shear. Explicit form of the stress function in the inhomogeneity as well as in the matrix was derived in their work. Then, a generalized and unified treatment was developed by Gong [3] for the elliptical inclusion embedded in an infinite matrix not only under the remote shear but also interacting by the screw dislocation. Besides, Shen et al. [4] developed a semi-analytical solution for the problem of an elliptical inclusion not perfectly bonded in an infinite matrix under anti-plane shear. Under the assumption of continuous tractions and discontinuous displacements across the interface, they used a model of a spring layer with thickness to simulate the interface. They found the non-uniform stress field and the average stresses in the inclusion is highly related to the aspect ratio of the inclusion and the parameter of interface simulation. For arbitrary distributed elliptical inclusions under remote shears, few works were found in literature.
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