13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 1 ˆ n u u i i i u u N u N u, u u N u (18) and the fluid pressure p similarly approximated by 1 ˆ n p p i i i p p N p N p, p p N p (19) where iu and ip are the nodal displacement and pressure, u iN and p iN are corresponding nodal displacement and fluid pressure shape functions. The crack opening displacement w is approximated by 1 ˆ n w w i i i w w N u N u, w w N u (20) where wiN are the appropriate crack opening displacement shape function. It will be shown later that the shape function wiN can be expressed in terms of the displacement shape functions u iN according to the relationship Eq. (14). Substitution of the displacement and pressure approximations (Eqs. (18) - (20)) and the constitutive equation (Eq. (3)) into Eq. (15) yields a system of algebraic equations for the discrete structural problem u 0 Ku Qp f (21) where T d K B DB , T T t u u u d dΓ f N b N t , T c w pdΓ Q N nN (22) By substituting Eqs. (19) and (20) into Eq. (17), the standard discretization applied to the weak form of the fluid flow equation leads to a system of algebraic equations for the discrete fluid flow problem p 0 Cu Hp f (23) where T T T c p wdΓ C Q N n N , T c p pdΓ H N k N , T c p p gdΓ f N (24) Then, the discrete governing equations for the coupled fluid-fracture problem can be expressed in matrix form as: u p 0 0 K -Q u f u C 0 0 H p f p (25) The above equations form the basis for the construction of a finite element which couples the fluid flow within the crack and crack propagation. 4. The extended finite element method and element implementation 4.1 Extended finite element approximation By adding special enriched shape functions in conjunction with additional degrees of freedom to the standard finite element approximation within the framework of partition of unity, the extended finite element method (XFEM) [16-17] overcomes the inherent drawbacks associated with use of the
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