13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Time-Domain BEM for Dynamic Crack Problems in Thin Piezoelectric Structures Hongjun Zhong1, Jun Lei1,*, Chuanzeng Zhang2 1 Department of Engineering Mechanics, Beijing University of Technology, 100124, China 2 Department of Civil Engineering, University of Siegen, D-57076 Siegen, Germany * Corresponding author: leijun@bjut.edu.cn Abstract A 2-D time-domain boundary element method (BEM) is developed to study the static and dynamic fracture problems in thin piezoelectric structures under electromechanical loadings. The traditional displacement boundary integral equations (BIEs) are applied on the external boundary and the hypersingular traction BIEs are applied on the crack faces. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the crack tip. The strongly singular and hypersingular integrals are evaluated by a regularization technique based on a suitable variable change. The nearly singular integrals arisen in thin structures are dealt with by two ways. The first one is based on a nonlinear coordinate transformation method for curve-quadratic elements. The second method is on an analytical integration method for straight quadratic elements to avoid the nearly singularity. A displacement extrapolation technique is used to determine the dynamic intensity factors (DIFs) including the dynamic stress intensity factors (DSIFs) and dynamic electrical displacement intensity factor (DEDIF). Some examples are presented to verify the effectiveness and stability of present BEM in thin piezoelectric structures. Keywords thin piezoelectric structure, time-domain boundary element method, nearly singular integration, dynamic intensity factors 1. Introduction With intrinsic electro-mechanical coupling characteristics, piezoelectric materials are widely used in smart structures. For typical engineering piezoelectric materials like PZT and PVDF, it’s difficult to be applied to the complicated shape structures. Therefore, painting technology is developed, which forms piezoelectric coatings on the structures to produce electro-mechanical coupling function. But cracks may occur in the coating or between the coating and the matrix during preparation or under complex electro-mechanical loadings. So it’s significant to study the fracture problems in the thin piezoelectric structure. For general dynamic crack problems in piezoelectric materials, numerical methods are more feasible due to the mathematical complexity of the initial boundary value problems. Particularly, Boundary Element Method (BEM) has certain advantages in fracture analysis. In 2008, time-domain BEM for transient dynamic crack analysis of linear piezoelectric solids was implemented by García-Sánchez et al. [1], who used the Lubich convolution quadrature formula for the temporal discretization and a collocation method for the spatial discretization. However, its extension to dynamic cracks in thin piezoelectric structures is not straight-forward, since the corresponding dynamic Green’s functions have quite complicated mathematical structures, which generate nearly singularity during integration when the field point is very close to the source point. In 2002, Liu and Fan [2] successfully applied the BEM in the static analysis of thin piezoelectric solids. The nearly singular integrals were solved by an analytical method. But they didn’t take cracks into consideration.
RkJQdWJsaXNoZXIy MjM0NDE=