13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- According to the singular integral equation theory [13], the solutions of Eq. (7) have the following form 1 1 2 2 1 1 2 2 1 2 ( ) ( ) ( ) , ( ) 1 1 F r F r f r f r r r = = − − , (14) where F1(r1) and F2(r2) are continuous and bounded functions. Once the solutions of the above integral equations are obtained, the TSIFs at the periodic crack tips can be computed from ( ) ( ) ( ) ( ) (1) (1)* 1 1 0 (2) (2)* 2 2 0 1 1 1 , 2 1 1 1 , 2 I I I I K a K F b E T b K a K F b E T b ν α π ν α π − = =− Δ − = =− Δ (15) where KI (1) and KI (2) denote the TSIFs at the tips of the long and short cracks, respectively, KI (1)* and KI (2)*are the corresponding nondimensional TSIFs, ΔT = Ta, and α0 is the coefficient of thermal expansion at x = 0. In Eq. (15), F1(1) and F2(1) have been normalized by (1+ν)α0ΔT. 4. Numerical Results and Discussion In the numerical examples, we use a graded system of alumina/silicon nitride (Al2O3/Si3N4) FGM for cutting tools applications to examine the effects of cooling rate on the TSIFs. The FGM is assumed to be a two-phase composite material with graded volume fractions of its constituent phases. The volume fraction of Si3N4 is assumed to follow a simple power function ( ) ( / )p V x x b = , (16) where p is the exponent determining the volume fraction profile. The material properties of the FGM are calculated using conventional micromechanics models [14] and the properties of Al2O3 and Si3N4 are given in Table 1 [15]. In the numerical calculations, we only consider the loading case of Tb = 0 (the initial temperature), which means that only the cracked surface x = 0 of the FGM plate is subjected to a temperature drop. Fig. 2a shows the normalized TSIF KI (1)* at the tips of the long cracks versus nondimensional time τ for various values of the cooling rate parameter τa. The crack spacing is h/b = 1, the length of the long cracks is a1/b = 0.1, the crack length ratio is a2/a1 = 0.2, and the material gradation profile index is p = 0.5. The TSIF under the sudden cooling condition ( τa = 0, and hence infinite cooling rate) is also included. For a given cooling rate (Ta/ τa), the TSIF initially increases with time, rapidly reaches the peak value and then decreases with time. The peak TSIF decreases dramatically with a decrease in the cooling rate (increasing τa). Moreover, the time at which the TSIF reaches its peak increases with a decrease in the cooling rate. Fig. 2b shows the normalized TSIF KI (2)* at the tips of the short cracks versus nondimensional time τ. All the geometrical and material gradation parameters are the same as those in Fig. 2a. The TSIF versus time response exhibits similar trend to that at the tips of the long cracks shown in Fig. 2a. Comparing the results in Figs. 2a and 2b, we can see that the peak TSIF at the short crack tip under the sudden cooling condition is about the same as the corresponding value for the long cracks. The peak TSIFs under finite cooling rates (nonzero τa), however, are significantly lower than the
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