4. DISCUSSION
In fabricating Ti/APC-2 nanocomposite laminates the superior and stable mechanical properties of the samples due to tensile tests should be maintained. Based on the rule of mixtures it is found see that the errors were very small between the predicted results and experimental data, i.e., the errors for strength were less than 7.90%, while the errors for stiffness less than 1.48%. That demonstrates the fabricated neat and nanocomposite Ti/APC-2 samples were of good quality and small scatterness after cyclic tests, please refer to 27.
Generally, it is reasonable to predict that the strength, stiffness and life of both neat and nanocomposite laminates, symmetrically and anti-symmetrically double-edge-cracked samples were lower for crack length 3.0 mm in the detrimental situations as illustrated in Tables 1(a) and 1(b) to 4(a) and 4(b). Similarly, the higher the inclined angles are, i.e., θ = 60° symmetric samples and θ = ±60° for anti-symmetric samples, the higher the mechanical properties and fatigue lives are. That is attributed to the longer distance between the two crack tips. It is also found the mechanical properties of anti-symmetric samples are slightly better than those of symmetric counterparts. The shortest distance, i.e., show cut, between the two crack tips for the anti-symmetric samples is slightly longer than that of symmetric counterparts.
The addition of nano-powder SiO2 in the APC-2 interfaces did improve the mechanical properties and cyclic lives, but not significantly. The enhancement of mechanical properties and lives was less than 10%. Some test data showed that the enhancement was even not occurred. It is reasonable to doubt that the enhancement at the crack tips was totally balanced by the crack tip stress intensity factors, i.e., the spreading of nano-powder was global; whilst the stress intensity factor as the crack tip was strongly a local point.
For symmetrically double-edge-cracked samples, as shown in Figure 1(b) and (c), the stress intensity factors, \(K_{I}\) and \(K_{\text{II}}\), in mixed mode can be expressed in Equations 1-3, where a is crack length, W denotes the width of sample, and B is thickness of FMLs. The critical value of SIF, \(K_{C}\), is simply adopted in Equation 4.
\(K_{I}=\frac{P}{B\sqrt{W}}f\left(\frac{a}{W}\right)\cos^{2}\theta\)(1)
\(K_{\text{II}}=\frac{P}{B\sqrt{W}}f\left(\frac{a}{W}\right)\text{cosθsinθ}\)(2)
where
\(f\left(\frac{a}{W}\right)=\frac{\sqrt{\frac{\text{πa}}{2W}}}{\sqrt{1-\frac{a}{W}}}\left[1.122-0.561\left(\frac{a}{W}\right)-0.205\left(\frac{a}{W}\right)^{2}+0.471\left(\frac{a}{W}\right)^{3}+0.190\left(\frac{a}{W}\right)^{4}\right]\)(3)
\(K_{C}=\sqrt{K_{I}^{2}+K_{\text{II}}^{2}}\) (4)
Using Equations (1)-(4) the results of \(K_{I}\) , \(K_{\text{II}}\) and\(K_{C}\) for two crack lengths at each inclined angle were easily obtained. In the consideration of the interaction of two crack tips of symmetrically double-edge-cracked samples the stress intensity factors should be corrected by Equations 5-6, where a * is the shortest distance between the midpoint of two cracks. The revised forms are
\(K_{I}^{0}\approx K_{I}\left[1+\frac{a^{*}}{2d^{2}}\left(2cos2\varnothing-cos4\varnothing\right)\right]\)(5)
\(K_{\text{II}}^{0}\approx K_{\text{II}}\left[1+\frac{a^{*}}{2d^{2}}\left(-sin2\varnothing+sin4\varnothing\right)\right]\)(6)
The results of stress intensity factors such as \(K_{I}^{0}\) and\(K_{\text{II}}^{0}\) were listed in Table 5. Similarly, refer to Figure 1(b) the stress intensity factors can also be easily received for neat and nanocomposite anti-symmetrically double-edge-cracked samples of two crack lengths at each inclined angle. According the interaction of both crack tips as shown in Figure 1(c) the revised results of stress intensity factors such as \(K_{I}^{0}\) and \(K_{\text{II}}^{0}\) for those samples can be calculated by using Equations 7-8, and the values are tabulated in Table 6, where a * = 1.0 mm for crack lengtha = 2.0 mm and a * = 1.5 mm for a = 3.0 mm.
\(K_{I}^{0}\approx K_{I}\left[1+\frac{{a^{*}}^{2}}{2d^{2}}\left(2cos2\varnothing-cos4\varnothing\right)\right]\)(7)
\(K_{\text{II}}^{0}\approx K_{\text{II}}\left[1+\frac{{a^{*}}^{2}}{2d^{2}}\left(-sin2\varnothing+sin4\varnothing\right)\right]\)(8)
It is interesting to see for symmetrically double-edge-cracked samples the revised stress intensity factor \(K_{I}^{0}\) decreases as the increasing of inclined angle, however, the revised \(K_{\text{II}}^{0}\)still keep the same value as 1.00\(K_{\text{II}}\), i.e., no change. As for anti-symmetrically double-edge-cracked samples both revised stress intensity factors, \(K_{I}^{0}\) and \(K_{\text{II}}^{0}\), decrease as the increasing of inclined angle; whilst, \(K_{I}^{0}\) reduces from 1.00160\(K_{\text{II}}\)for a = 2.0 mm, and 1.00421\(K_{I}\) fora = 3.0 mm at inclined angle 30°, i.e., \(\backslash nK_{I}^{0}\)is over 1.0\(K_{I}\) at θ = 30°, down to 0.99954\(K_{I}\) fora = 2.0 mm and 0.99890\(K_{I}\) for a = 3.0 mm at θ = 60°, i.e., \(K_{I}^{0}\) is lower than 1.0\(K_{I}\).
Based on the suitable methodology to predict crack growth rate and life the well-known Paris Law has to be modified. In the mixed mode fracture the effective revised stress intensity factor range, \(K_{e}\), instead of \(K_{I}^{0}\) and \(K_{\text{II}}^{0}\), the hybrid FMLs material properties and mechanical behavior, and most importantly the fracture process and mechanisms need be involved.
The surprising and interesting phenomenon is the occurrence of a small broken ellipse at the center part when both side crack tips grow towards each other along the shortest path to deviate slightly and do not coalesce. Due to their interaction, both crack tips run around each other in a certain distance until, at some later instant, each of them merges with the other crack. The failure mechanisms of nanocomposite symmetrically double-edge-cracked samples due to cyclic loading were shown in Figure 6(a) the enlarged scheme and Figure 6 (b) the photo of failed sample. Similarly, the failure mechanisms of nanocomposite anti-symmetrically double-edge-cracked samples due to cyclic loading were presented in Figure 7(a) the enlarged scheme and Figure 7(b) the photos of three failed samples.
From on the above-mentioned the particular failure mechanisms such as the small elliptical pieces formed at failure in both kinds of FMLs that the irregular crack growth rate and path make the fatigue life prediction a very complicated task.