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.