13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Real and portrait displacement discontinuities in half-plane y 0, with a traction-free surface (after Crouch and Starfield [25]). In the present study, in order to obtain high precision, quadratic collocation displacement discontinuity modified for half plane crack problems with a traction free surface are used. Dk(ξ) is equation that can be used to calculate two fundamental variables of each element (the opening displacement Dy and sliding displacement Dx). D k x y D k i n i i j , , ( ) ( ) 1 (3) Where (,. and ) 1 1 1 y x kD i eD D , (.. and ) 2 2 2 y x kD ieD D , (.. and ) 3 3 3 y x kD ieD D are the quadratic displacement discontinuities, and using the equal length of each sub element 3 2 1a a a ) /(16 ) ( ) (9 9 ), ) /(16 ( ) (9 9 ), ) /(48 3 ( ) (3 3 1 3 2 1 2 1 3 1 3 3 1 3 2 1 2 1 3 1 2 3 1 3 2 1 2 1 3 1 1 a b a a a a a a a a a a (4) are their quadratic element shape functions. In Fig. 2, A quadratic displacement discontinuity (DD) element is divided into three equal sub-elements (each sub-element contains a central node for which the nodal displacement discontinuities are evaluated numerically) [26]. DX=-Dx |Fy| |Fx| y - |Fx| Dy x y x' y Dy= Dy DX x 0 yx xt
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