13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 0 N T and 0 %N T are the initial and current normal cohesive strength, the subscript (H) represents that the initial cohesive strength is dependent on the triaxiality condition. Nk and % Nk are the initial and current normal stiffness of the cohesive element. 1δ is a parameter relating to the critical separation δ0, the meaning is the same as in Equation (1). For the triaxiality dependent cyclic CZM, it is difficult to judge the loading and unloading conditions because the separation δ and the critical separation δ0 of the cohesive element alter simultaneously. So, a relative normal separation and its increment are defined by Equation (5) and Equation (6): ( ) 0 δ δ δ = N N re N H (5) ( ) ( ) ( ) ( ) 0 0 δ δ δ δ δ δ δ −Δ −Δ −Δ Δ = − = − t t t N N N N N t t t re re re t t t N N H H (6) δN re and δ N are the relative and absolute normal separation, respectively. 0δ N is the critical normal separation, the subscript (H) represents that it is dependent on the triaxiality condition. δΔ N re is the relative normal separation increment. The subscripts t-Δt and t stand for the previous and current calculation time increment. δ−Δ Nt t and δ Nt , ( ) 0δ −Δ t t N H and ( ) 0δ t N H , ( ) δ −Δ N re t t and ( ) δ N re t correspond to the absolute normal separation, the critical normal separation and the relative normal separation in the time increment t-Δt and t, respectively. If in the different time increment the triaxiality condition does not change, the two critical separations ( ) 0δ −Δ t t N H and ( ) 0δ t N H should be the same. A loading step is indicated by δ Δ N re >0, an unloading step is indicated by δ Δ N re <0. At the first loading step, the traction separation response goes along the monotonic TSL’s path (Equation (1)). Afterwards the unloading and reloading process is defined by Equation (7). ( ) 0 δ δ −Δ ⎡ ⎤ = + Δ ∗ ⎣ ⎦ % t N N N N t t t N re H T T k (7) −Δ N t t T and N tT are the normal traction in the time increment t-Δt and t. The meanings for the other symbols can be checked in the previous paragraphs. In every loading process a maximum relative normal separation max δ N re may be reached. During the reloading process, when the current relative normal separation exceeds this maximum value, Equation (8) is used for the new path. ( ) ( ) 2 1 0 1 1 0 1 0 0 2 δ δ δ δ δ δ δ δ δ δ ⎧ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎪ ⎢ ⎥ − ≤ ⎜ ⎟ ⎜ ⎟ ⎪ ⎢ ⎥ ⎣ ⎝ ⎠ ⎝ ⎠ ⎦ = ⎨ ⎪ > ⎪ ⎩ % % N N N N re N H N N N re N H T T T (8) In the unloading process, it is still an open topic how the triaxiality influences the cohesive parameters. In this paper, a simple assumption is made. The triaxiality value keeps constant in the unloading process and a part of the reloading process. The definition about this part of the reloading
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