13th International Conference on Fracture June 16–21, 2013, Beijing, China 5 described in the incremental form as Δ = ( ) T F D with c f u f q D D F D Δ + Δ −Δ − = ) ( (1 ) ( ) σ and c f q σ σ = . (9) The equation above indicates that the unloading/reloading stiffness the cohesive zone, F(D), varies with D from σc /Δc for no damage (D=0) to zero for D=1. The total damage in terms of a loading history should be the sum of the monotonic damage and cyclic damage D= Ds + Dc = Δ Δu −Δf + Dc (10) 3.2. Implementation of cohesive element into ABAQUS The commercial software ABAQUS[18] allows the user to define a new element conveniently, via the interface UEL. The formulation of the cohesive element is referred to the ABAQUS theoretical manual. We have to provide the cohesive element stiffness, the residual nodal force vector and derive the cohesive traction and displacement jump according to the cohesive law. These variables are needed to calculate the virtual work generated by the cohesive zone such that the internal work equals to the external virtual work for every kinematically admissible displacement field in accordance with Eq. (1). As illustrated in Figure 4, a four node cohesive element is defined in two dimensional x-y coordinates. The nodal displacement vector corresponding to 4 nodes is defined by [ ] 1, 2,3, 4 , T = = i u vi i u (11) with u and v as nodal displacements in x and y direction, respectively. Then the local displacement jump vectors Δ are written using the shape functions and the nodal displacement as = = s n ⎡Δ ⎤ ⎢ Δ ⎥ ⎣ ⎦ Δ αBu (12) with B as the matrix about the shape functions, 1 2 2 1 1 2 2 1 - 0 - 0 0 0 = 0 - 0 - 0 - 0 N N N N N N N N ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ B , (13) where N1=(1-s)/2 and N2=(1+s)/2 are linear interpolation functions in the intrinsic coordinate system and α is the rotation matrix denoted by cos sin sin cos θ θ θ θ ⎡ ⎤ =⎢ ⎥ ⎣ − ⎦ α . (14) The cohesive element stiffness matrix and the traction force term are given by Ke= BTDB e∫ dA= le 2 BTDB -1 1 ∫ ds, Te= BT e∫ TdA= le 2 BTT -1 1 ∫ ds, (15) where le is the length of the element , T is the traction vector based on the Eq. (2) and D is the
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