13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) ( ) ( ) ⎩ ⎨ ⎧ − ⋅ < − ⋅ ≥ =− = = 0 0 1 1 * * x x n x x n x H (11) Where, x* is the closest point on the crack surface for point x. The unit normal vector n is defined in Figure 1. The vector functions g(r,θ) are defined as ( ) ( ) {()()()()} ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = = = θ θ θ θ θ θ θ θ θ θ θ sin 2 , cos sin 2 , sin 2 , cos 2 sin , , , , , , , , 4 3 2 1 r r r r g r g r g r g r r g x g (12) Where, (r,θ) are the local polar coordinates of point x. The local polar coordinate system is established at the crack tip in 2D case [1] or crack front in 3D case [7]. In (12), the first function is discontinuous across the discontinuity surface whereas the other three functions are continuous. So the displacement jump across the discontinuity surface comes from the generalized Heaviside function (11) and the first function in (12). The equilibrium equations are obtained after the discretization of weak form of equilibrium equations and the implementation of the contact constrain by Lagrangian multiplier method. ( ) Ku f F P c = + (13) Where, u is the vector of nodal displacements including the conventional DOFs and additional nodal enrichment DOFs. K is the stiffness matrix. f is the nodal load vector resulted from body force and prescribed traction on σΓ . Fc is the equivalent nodal force vector resulted from the contact forces P acting on the contact surfaces which are equivalent to the contact stress acting on the crack surface. Note that the unknown vectors in Eq.(13) are u and P. 3.2. The derivation of equivalent nodal force resulted from the contact forces on the crack surfaces in the XFEM formulation In the case that the contact surfaces are embedded in the interior of element, the contact pairs can be easily formed at the positions where the crack line/surface intersects the edges of the element. The two contact points in one contact pair have the same coordinates as the intersection point. Let cx be the position of any contact pair. Then the displacement jump can be computed according to (10)~(12) and obtained as ( ) [ ] [ ] ( ) ( ) ( ) ( ) ∑ ∑ = = + = − = nd k k c k nd j j c Hj c c c r N N 1 1 1 2 1 2 2 x c b x u x u x u x (14) Where, 1 cx and 2 cx are two contact points in the contact pair located on crack surface 1 cΓ and 2 cΓ respectively. Because no initial gap exists between the two contact surfaces, c c cx x x = = 1 1 . The equivalent nodal forces Fc can be obtained from virtual work done by the contact forces P acting at the contact points. It is assumed that the element is cut by contact surface into two pieces, as shown in Figure 2. In order to clarify the following procedure for more general cases, the nodes
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