13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- definitions of ( ) P r i a , ( ) P r i b and λ are given as follows. i a i a i aP r P r du = − Δ ( ) (6a) i b i b i bP r P r du = − Δ ( ) (6b) { } ( ) ( ) ( ) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + = ,1 min min ,0 2 2 P P r P r i b i a i n μ λ (7) Where, i a du Δ and i b du Δ are tangential incremental relative displacements in local a and b directions respectively. The incremental relative displacements in global and local coordinate system for i-th contact pair at current load step are defined as ( ) j n a b or x y z du du dui j i j i j , , , , , 2 1 = Δ = − (8) The normal relative displacement for i-th contact pair at current load step is defined as ( )2 1 i n i n i n i nu u du du Δ =Δ + − (9) Where, i nuΔ is the initial normal relative displacements at current load step. 3. Discretization by XFEM formulation The XFEM is a promising method to simulate the existence and growth of the discontinuities, such as cracks, without the need to make the mesh conforming it. In order to characterize the discontinuous displacement field resulted from the embedded discontinuity, the Heaviside functions or asymptotic crack tip functions are often used to enrich standard continuous displacement fields of the elements cut entirely by the discontinuity surface or those including the crack tip in 2D or crack front in 3D case, respectively. The additional nodal degrees of freedom corresponding to these enrichment functions are needed. 3.1. The XFEM approximation In XFEM, the displacement approximation at an arbitrary point x in the element with embeded discontinuity takes the form [ ] [ ] ( ) ( ) ∑ ∑ ∑ ∑ = = = = − + − + = = + nd k l l k k l kl k nd j j j Hj nd i i i g r g r N N H H N 1 4 1 1 1 ( , ) ( , ) ( ) ( ) ( ) ( ) ( ) θ θ c x x u b u x u x u x (10) Where, nd is the number of nodes in one element, iu is the continuous displacement field at node i. The last two terms form the discontinuous part [[ ]] u . H(x) is generalized Heaviside function, and bHj is the vector of additional translational DOFs corresponding to generalized Heaviside function. gl (l=1,2,3,4) are the tip enrichment functions, and ckl is the vector of additional translational DOFs related to the l-th tip branch function for node k. The generalized Heaviside function is defined as
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