*2.3. Crack Growth Analysis*

The direction of the crack path under linear elastic conditions must be computed to facilitate crack propagation simulation. The maximum circumferential stress theory states that for isotropic materials under mixed loading mode the crack grows in a direction normal to a maximum tangential tensile stress. The tangential stress is estimated in polar coordinates as 

$$\sigma\_{\theta} = \frac{1}{\sqrt{2\pi r}} \cos\frac{\theta}{2} \left[ K\_I \cos^2\frac{\theta}{2} - \frac{3}{2} K\_{II} \sin\theta \right] \tag{14}$$

The direction normal to the tangential maximum stress can be obtained by resolving *dσθ*/*dθ* = 0 for *θ*. The nontrivial solution is determined by

$$K\_I \sin \theta + K\_{II} (3 \cos \theta - 1) = 0 \tag{15}$$

which can be solved as

$$\theta = \pm \cos^{-1} \left\{ \frac{3K\_{II}^2 + K\_I \sqrt{K\_I^2 + 8K\_{II}^2}}{K\_I^2 + 9K\_{II}^2} \right\} \tag{16}$$

The sign of *θ* must be opposite the sign of *KI I* to ensure the optimal opening stress associated with the crack direction [28]. Figure 3 illustrated the two possibilities of the crack growth direction.

**Figure 3.** Sign of the crack growth angle.

In the case of fatigue crack growth, the resulting stress intensity range at each crack tip must exceed the stress intensity threshold, specified as

$$
\Delta \mathcal{K}\_{th} = f \Delta \sigma\_{th} \sqrt{\pi a} \tag{17}
$$

where *f* is a geometrical and loading function and Δ*σth* is the stress range limit. According to Equation (17), the crack is not propagated if Δ*σ* < Δ*σth*. This equation was practically modified by using another parameter known as the equivalent stress intensity factor range, <sup>Δ</sup>*KIeq*. Therefore, if Δ*KIeq* > Δ*Kth*, this indicates commencement of fatigue crack growth. This parameter is set to

$$
\Delta K\_{Icq} = \Delta K\_I \cos^3(\theta/2) - 3\Delta K\_{II} \cos^2(\theta/2)\sin(\theta/2) \tag{18}
$$

In the modified equation of the Paris law, Tanaka [20] derived an innovative law known as the power law for determining crack growth in response to fatigue with the equivalent stress intensity factor (Δ*Keq*) as

$$\frac{da}{dN} = \mathcal{C}(\Delta K\_{\text{eq}})^m \tag{19}$$

where *a* is the length of the crack, *N* is the number of cycles, *C* is the Paris constant (mm/cycle), and *m* is the Paris exponent.

The total number of fatigue lifecycles can be calculated using Equation (19) for an increase in crack length as

$$\int\_{0}^{\Delta a} \frac{da}{\mathcal{C}(\Delta K\_{eq})^{\text{M}}} = \int\_{0}^{\Delta N} dN = \Delta N \tag{20}$$
