13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- To apply the time-domain BEM to the crack analysis of thin piezoelectric structures, two methods are developed to deal with the nearly singular integrals. 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. Numerical results are presented to verify the accuracy of the present integral equations and the time-domain BEM. 2. Time-domain BIEs for Piezoelectric Materials 2.1. Piezoelectric Equations Consider a 2D homogeneous, linear cracked piezoelectric solid occupying domain with boundary . To describe the electric-elastic fields, the ‘extended’ variables uJ, fJ, σiJ, εJi are defined: j J u u , 1,2 3 j J f J j f q J , ij iJ iD , 1,2 3 ji Ji i J j E J , (1) which combine the elastic variables involving the displacement uj, body force fj, stress σij, strain εij, and the electric ones including electric potential , charge q, electrical displacement Dj, field Ej. Without body forces and electrical charge, the governing equations and the constitutive equations under quasi-electrostatic assumption are given by * , iJ i JK Ku , iJ iJKl Kl C , (2) where ρ is the mass density, “,” designates spatial derivative, while “ ” denotes temporal derivative. The capital index is from 1 to 3 while lower case letter index takes 1 or 2. The extended Kronecker delta * JK is defined by * , 1,2 0 otherwise JK JK J K . (3) Material constants CiJKl are as follows, , 1,2; 1,2 , 3; 1,2 , 3; 1,2 , 3 ijkl jil iJKl kli il c J j K k e K J j C e J K k J K . (4) in which cijkl, eijk and κik are the elasticity constants, piezoelectric constants and dielectric constants, respectively. The extended strain and displacement relations are given by , , / 2 ij i j j i u u , , i i E . (5) 2.2. Time-domain Boundary Integral Equations On the boundary , the displacement and traction BIEs are G G G 0 0 ( ) , d d C IJ J IJ J IJ J IJ J c u t u p p u p u x x , 0 x , (6) G G G 0, d d C J IJ J IJ J IJ J p t d p s u s u x , 0 C x , (7) where cIJ(x) are free term constants, “” denotes Riemann convolution, Ju are the extended crack-opening-displacements, GIJ u and GIJ p are 2D time-domain dynamic displacement and the
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