13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- vector to the boundary. A complex variable is introduced as 0 0 0 ( ) M M M M z z x x y y . (16) With this transformation, 1 e M q M M I d , 2 1 e M q M M I d . (17) q can be considered as a function of M , and its first Taylor series at 0 M is 0 ( ) ( 0) O( ) O( ) q M q M M q M . (18) With the substitution of Eq. (18) into MI , 0 0 1 1 ( ) e e M q q M q M M M I d d . (19) The first part in Eq. (19) can be calculated by Gaussian quadrature formula. The second integral is strongly singular but can be evaluated analytically as 1 log( ) | e e M M M d . (20) For MI , the Taylor-series is expanded to two terms. Then, the integral can be transformed to 0 0 0 0 2 2 ( ) 1 1 e e e q q q M M M q M q M M M M I d d d . (21) The first integral can be computed by Gaussian integral quadrature. The second and third integral can be evaluated analytically as 2 1 1 e e M M M d , 1 log( ) | e e M M M d . (22) 4. Computation of nearly singular integrals When the collocation point is very close to the integration point in thin piezoelectric structures, nearly singular integrals appear. To deal with this problem, two methods are developed for curve quadratic elements and straight quadratic element. On the analogy of the solution to the singular problems in section 3, the nearly singular integrals can be classified into three kinds 1 2 0 Γ ( ) ( ) Γ e M M q M M n n J d z z x x , 1 2 0 2 Γ ( ) ( ) Γ ( ) e M M q M M n n J d z z x x , 1 0 2 0 0 Γ ( ) ( ) Γ e M M q M M n n J d z z x x .(23) 4.1. Curve quadratic element For isoparametric quadratic element, similar method as the singular integrals can be applied. Local coordinate system of [ 1,1] is introduced. When collocation point 0x is very close to e , the nearest point on e to 0x is assumed to be ( , ) x y Q which has the local coordinate , so M M z x y . (24) Let M M M z z , 0 0 M M M z z , so MJ can be transformed into 0 Γ 1 e M q M M M J d . (25) With the substitution of Taylor expansion of q, MJ can be separated into two parts as follows
RkJQdWJsaXNoZXIy MjM0NDE=