13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Local coordinate system on the contact surface 2 cΓ Then the normal gap nuΔ between a contact pair can be defined as ( ) ( ( ) ( )) x u x u x n⋅ Δ = − 2 1 c c n c u (3) Where, ( ) 1 c u x and ( )2 c u x are the displacements of contact points on 1 cΓ and 2 cΓ respectively. Since part of the system energy will be dissipated due to the friction, the frictional contact problem is nonlinear and path-dependent and increment solution strategy is needed. So at every load step, when frictional sliding takes place between two contact surfaces, relative incremental sliding in tangential directions are defined as follows. ( ) ( ) ( ) du x du x a⋅ − Δ = 2 1 c c a du (4a) ( ) ( ) ( ) du x du x b⋅ − Δ = 2 1 c c b du (4b) Where, du is the vector of incremental displacement at current load step. The contact conditions in normal and tangential directions proposed by Christensen et al. [5] are expressed as a B-differentiable equation set. Due to the assumption of small displacement and small strain, the discrete point-to-point contact model is employed in the following, so that the contact conditions will be formulated by the quantities at these discrete contact pairs. For i-th contact pair, they are written as ( ) { } , 0 min , H2 = = Δ i n i n i i c i r u P du dP (5a) ( ) ( ) 0 , H3 = = − P P r i a i a i i c i λ du dP (5b) ( ) ( ) 0 , H4 = = − P P r i b i b i i c i λ du dP (5c) Where, i c du is the incremental displacement vector of two contact points consisting of the i-th contact pair. i dP is the vector of increment contact force acting at the i-th contact pair. i nP , i aP and i bP are the total contact forces in the normal and two tangential directions. r is a positive scalar. i nuΔ is the normal gap. Eq. (5a) corresponds to the normal contact condition, Eqs. (5b) and (5c) correspond to the tangential contact conditions in local a and b directions respectively. The
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