13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 0 0 ln , i i i i i v u x y u （17） It means that the cover function of the nodei is interpolated in terms of nodal displacements( , ) 0 0i iu v . 2.2 Cover interpolation for the nodes located on the essential boundary For a typical node j located on the essential boundary, suppose that thereMnodes in the support of cover j C (Fig. 1). At the local cover j C , a discrete error norm is defined as M k k j k j j lb j u u 1 0 ( ) ( ) ( ) x x u Φ x （18） where , , , 1 [ 1 1 j j j j j j j j j x x x x x Φ ] , j j M j M x x （19） j x is the coordinate of node j . Similarly, the y-displacement interpolation ( )x lb j v is given by M k k j k j j lb j v v 1 0 ( ) ( ) ( ) x x V Φ x （20） where 0 20 10 0 M j v v v V （21） Eq. (18) and (20) can be rewritten in the form j j lb j lb j lb j v u D Φ x x u x * （22） where j M j j M j j Φ Φ Φ Φ 0 0 0 0 1 1 * Φ （23） 0 0 20 20 10 10 M M j u u u u u u D （24） Substituting Eq. (14) and (22) into Eq. (2) leads to n i l i i h u u 1 0 x x x 2 1 1 1 0 1 ln 0 n j M k k j k j n i i i u Φ u x x x x （25） where 2 1 n n n is the total number of neighboring nodes where the position point xlocated in the cover support, 1n denotes the node set not on the essential boundary, 2n denotes the node set located on the essential boundary. The MSIM possesses the delta property and preserves the completeness of the field up to the order of the basis. The proof of these properties can be referred to [11]. 3. The application of virtual crack closure technique in MSIM Virtual crack closure technique (VCCT) is proposed by Rybicki and Kanninen [10] mainly dealing with two dimensional problems. In this method, the strain energy release rate for the crack propagation is always calculated based on the FEM analysis results. Here, taking a finite crack body as an example, the basic concept of virtual crack closure technique will be briefly introduced. The length of crack is a, and the propagation length is ∆ܽ . When the quadrangle mesh is employed,
RkJQdWJsaXNoZXIy MjM0NDE=