13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- process is like Equation (8). When the current relative normal separation is less than δN re , the triaxiality value keeps constant. Once δN re is exceeded, the triaxiality value updates again. After the cohesive element is broken, in the unloading or compression period in order to avoid the adjacent continuum element penetrating each other, a contact condition should be considered. Here, when δN <0, Equation (9) is used. δ = N N N T k (9) A new damage evolution equation is proposed here, as Equation (10) ( ) ( ) ( ) ( ) 1 0 0 0 1 λ σ δ − ⎛ ⎞ ⎡ ⎤ ⎜ ⎟ ⎢ ⎥ = Δ − ≥ ⎜ ⎟ ⎢ ⎥ ⎡ ⎤ − ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ & & m H N H N e re N N H H A e T D D H T T D (10) D is the damage variable. &D is the damage evolution. It is always a positive value. ( ) 0 N H T is the triaxiality dependent normal cohesive strength. NT is the cohesive traction. δ Δ N re is the relative normal separation increment. σe is the material endurance limit. H is the triaxiality value. A and m are the material dependent damage controlling parameters. λ is the material dependent triaxiality influence coefficient. A schematic illustration for the cyclic process of the CZM is shown in Figure 2. The new cyclic CZM is contained in a user defined subroutine, and it is connected with the commercial finite element program ABAQUS. The fatigue damage is calculated for every time increment. load cycles 0 a b c F one cycle NT 0 a c b max NT time t-Δt time t 1 1 δN re max δ N re ( ) δ N re t ( ) δ −Δ N re t t N tT −Δ N t t T σe ( ) 0δ N N H k ( ) 0δ % N N H k Figure 2. Schematic illustration for cyclic process of the CZM 4. Numerical application The triaxiality dependent cyclic CZM is applied for fatigue simulation. For triaxiality dependent behavior, some results from literature [20] are used. In reference [20], the examined material is S460N. For convenience, material S460N is chosen for the whole investigations in this paper, too. Some basic mechanical properties for material S460N are shown in Table 1. Table 1. Mechanical properties for material S460N
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