13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- conventional finite element methods and enables the crack to be represented without explicitly meshing crack surfaces, and so the crack geometry is completely independent of the mesh and remeshing is not required, allowing for the convenient simulation of the fracture propagation. The XFEM approximation of the displacement field for the crack problem can be expressed as [17] 4 1 , , cr tip l l l I I I I I I I I I I N I N I N l H H B r B r u x x u x x x a x b N N N (26) where N is the set of all nodes in the mesh, cr N the set of nodes whose support are bisected by the crack surface cΓ , tip N the set of nodes whose support are partially cut by the crack surface, I x N and I x N are the standard finite element shape functions, Iu are displacement nodal degrees of freedom, Ia and l Ib are additional degrees of freedom for the displacement, and H x and , l B r are the appropriate enrichment basis functions which are localized by I x N . The shape function I x N can differ from I x N . The discontinuity in the displacement field given by a crack cΓ can be represented by the generalized Heaviside step function 1 0 1 0 d H H d sign d d x x x x x (27) where d x is the signed distance of the point x to cΓ . The enrichment basis functions , l B r are required to model the displacement around the crack tip, which are generally chosen as a basis that approximately spans the two-dimensional plane strain asymptotic crack tip fields in the linear elastic fracture mechanics: 4 1 sin 2 cos 2 sin 2 sin cos 2 sin l l B r (28) where ,r are the local polar coordinates at the crack tip. The first function in Eq. (28), sin 2 , is discontinuous across the crack faces , so the Heaviside enrichment in Eq. (26) can be removed in the displacement field approximation of the elements which are partially cut by the crack surface. According to Eq. (26), the displacement discontinuity between the two surfaces of the crack can be obtained as 1 1 2 2 , cr tip I I I I I N I N B r w x u x u x x a x b N N cΓx (29) Combination of Eqs. (29) and (20) enables determining the shape function wN . The fluid pressure field within the crack is approximated by cr p I I I N p N p x x cΓx (30) where p IN x are the standard finite element shape functions. In some cases, it can also be chosen as a special function so as to allow for the pressure singularity at the crack tip and the associated near-tip asymptotic fracture opening associated with a zero-lag viscosity dominated regime in a
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