13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 4.4. Comparison of the MD results for stress singularity and angular function The order of stress singularity λ is obtained for various angle ω using Eq. (5). The angle ω is varied from 90° to 180°. The value of λ is shown in Fig. 13. A complex value of λ appears in the range 174° < ω ≤ 180°. In the case of ω = 180°, λ is 0.5±i ε (where ε is a constant that depends on the material combination: 0.0042(Cu-Au)). This case corresponds to a notch. Figure 14 shows both log plots of the data shown in Fig. 12. Figures 14(a) and 14(b) are stress distributions including and subtracting the initial stress, respectively. As shown in these figures, the stress distribution after subtracting the initial stress is straighter than that before subtracting the initial stress. Next, the results for ω = 170° are precisely investigated. Figure 13 shows that the imaginary part of λ for ω = 170° does not exist and that three roots exist corresponding to λΙ = 0.499, λII = 0.462, and λIII = 0.461. The angular functions are continuous at the interface, and f II and fIII are approximately 0 around θ = 0° and at the free surface ( θ = ±170°). Hence, the stress distribution along the interface may be expressed by a power-law with an index of -0.499. The stress distributions shown in Figs. 8 and 10 are then compared with the stress distributions approximated using !yy =Kyyr "0.499 . Here, r is used in place of x. Both log plots for σyy in the incoherent model are shown in Fig. 15. Figure 15(a) represents the stress distribution before subtracting the initial stress due to surface stress. Figure 15(b) demonstrates the stress distribution after subtracting the initial stress. The blue line in these figures indicates the approximated stress distribution using Eq. (9). As shown in Fig. 15(b), the atomic stress σyy with the initial stress subtracted agrees fairly well with the approximated line in 0.6 nm < r < 2 nm. However, the atomic stress increases near the tip and is larger than the approximated line given by the power-law that is derived from the theory of anisotropic elasticity. This might be due to a variation in surface stress near the tip. Log plots of atomic stress σyy in the coherent model are shown in Fig. 16. Figure 16(a) shows the distribution of atomic stress σyy corresponding to Fig. 15(a). The blue line indicates the plot of the same line shown in Fig. 15. The blue line does not agree with the atomic stress shown in Fig. 16(a). The stress distribution with the initial stress subtracted agrees fairly well with the blue line, as shown in Fig. 16(b). However, the atomic stress σyy becomes larger than the value estimated from the theory of anisotropic elasticity near the tip. Here, the order of stress singularity is determined as a function of distance along the interface using eqs.(2)-(5). The interface stress and interface elastic moduli are obtained as functions of the distance of the wedge tip along the interface. These relationships are used for solving eq.(5). The order of stress singularity is plotted against the distance as shown in Fig. 17. The plotted values are approximated using the power law function of the distance r as !1 r( ) =0.496+0.00114 r ! ( ) !4.45 . Here, !=1nm. Then, the distribution of atomic stress along the interface is presented in Fig. 18. It is found that the atomic stress can be expressed by the equation of 4.8r! "1 r( ) considering the interface stress and elasticity.
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