13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- temperature of 300K, constantly through re-calibration of the speed of atoms. Ladder-shaped loading is applied on the boundary, as shown in Fig.8. From Fig.9, the traction initially increases as the crack opening distance was increased until they reached a peak value, then decreased as the crack opening was further increased. To represent the CZM, a model that includes an exponential term is suggested to approximate the traction-separation response for tensile mode failure. The complex mechanisms of deformation, i.e., phase transition, dislocation, and crack blunting, were contained in the traction-separation (T-S) law. A in the T-S curve indicated damage occurred, which was shown by the typically non-linear decreasing of the curve. It corresponded to remarkable dislocations emission in the crack tip. B in the curve corresponds with the moment at which phase transition occurs in the MD model. The curve shows a wide scatter band in the region beyond the elastic deformations, since the data contain traction and separation in all stages of atoms along the crack path. During initial crack growth the scatter band in the cohesive law is very large. This behavior implies that the initial crack starts to grow at different maximum tractions and is not in a steady-state. With crack propagation, the scatter band becomes narrow, which indicates that crack growth becomes steady[15]. 0 2 4 6 8 10 12 0 5 10 15 20 25 σyy(GPa) Crack opening displacement(Å) T-S data point smooth curve A B Dislocation Phase Transition HCP BCC Figure 9. Traction-Separation relationship for α-Ti under tensile loading A relationship between the traction and the crack opening displacement must be obtained when we use cohesive element. In the CZM, the fracture process zone is simplified as being an initially zero-thickness zone, composed of two coinciding surfaces. Under loading, the two surfaces separate and the traction between them varies in accordance with a specified T-S law. In our study a parameterized exponential T-S law is proposed based on the MD simulations. The traction-separation law (F( λ)) is then implemented through Eq.(5). The non-dimensional parameter ( λ) in Eq. (5) relates the normal (un) separation to the maximum allowable normal ( δn) separation of the cohesive element[14]. In Eq.(5) the maximum cohesive strength ( σ max) is simulated at room temperature. The constants in Eq.(5) are obtained in table 2. According to Eq.(5) and table 2, the fitting curve of traction-separation relationship has been illustrated in Fig.10.
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