13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 4.2. Straight element For isoparametric quadratic element, the denominators of the nearly singular integrals possess a relative high order of , which brings difficulty for the integration. So the non-isoparametric quadratic straight line element is introduced. The boundary quantities are described by the same shape functions as the isoparametric quadratic element, while the geometry quantities x and y can be described as 3 1 1 3 2 2 x x x x x , 3 1 1 3 2 2 y y y y y , (36) in which 1 1 ( , ) x y and 3 3 ( , ) x y are respectively the start and end point of the element. With the above expressions, 0 M M z z can be represented by a linear function of , 0 0 0 ( ) ( ) ( ) M M M M M z z a c b d x y , (37) where 3 1 2 x x a , 1 3 2 x x b , 3 1 2 y y c , 1 3 2 y y d . So gains the following expression, 0 0 0 [ ( )] M M M M z z b x d y a c . (38) Nearly singular integrals can be expressed as 0 Γ e q M M M M L dz z z , 0 2 Γ ( ) e q M M M M L dz z z . (39) 2 2 1 0 2 0 0 Γ [ ( ) ( )] e q M M M M M M n n a c L dz A z z x x . (40) As the difference between ML and ML is just a coefficient, only Eq. (39) should be evaluated. Let 1 q , with the substitution of Eq. (38) into 1, the shape functions can be expressed as 0 2 0 2 3 2 3 1 2 1 2 1 3 ( ) [2 ( ) ]( ) ( )( ) ( )( ) M M M M M M M M M M M z z B A z z B A B A A , (41) where M M A a c , 0 0 ( ) M M B b x d y . Therefore ML can be separated into three parts as follows, 0 2 3 1 0 2 2 Γ Γ Γ 1 2 1 3 1 2 1 3 2 3 2 0 Γ 1 2 1 3 2 ( ) 1 ( ) ( )( ) ( )( ) ( )( ) 1 + ( )( ) e e e e M M M M M M M M M M M M M M M M M M M B A dz z z dz dz z z A A B A B A dz A z z . (42) The three integrals can be evaluated analytically, 0 2 0 Γ ( ) ( ) 2 e e M M M M M z z z z dz , (43) Γ e e M M dz z , 0 0 Γ 1 log( ) e e M M M M M dz z z z z (44) With similarity of the expressions, the integrals can be evaluated in the same way when 2,3 q . Similarly, ML can also be divided into three parts. Two of them can be integrated by Eq. (44), the other one can be evaluated as follows,
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