**2. Mechanical aspects of fatigue fracture**

Analysis of stress and strains for cyclic loading is needed for dealing with engineering situation such as vibratory loading which lead the component to fatigue fracture. In some practical applications the material operates at a maximum and minimum stress levels that are constant. This is known as constant amplitude stressing and is shown in **Figure 3**.

The stress range is the difference between the maximum and minimum stress values, Δ*σ* ¼ *σ max* � *σ min* , the mean stress, *σ<sup>m</sup>* is the average of maximum and

**Figure 3.**

*Various type of cyclic loadings. a) Completely reversed stressing, (b) nonzero mean stress, (c) zero to tension stressing.*

minimum of stress values that may be zero (**Figure 3a**). Half range is named stress amplitude. Mathematical expressions are as follows [1]:

$$
\sigma\_a = \frac{\sigma\_{\text{max}} - \sigma\_{\text{min}}}{2} \tag{1}
$$

$$
\sigma\_m = \frac{\sigma\_{\text{max}} + \sigma\_{\text{min}}}{2} \tag{2}
$$

The stress ratio R and amplitude ratio A is defined as:

$$
\sigma\_d = \frac{\sigma\_{\min}}{\sigma\_{\max}} \tag{3}
$$

$$A = \frac{\sigma\_d}{\sigma\_m} \tag{4}$$

#### **2.1 S-N curves**

Stresses (S) versus life (N) is an engineering representative of fatigue behavior of materials. When we do cyclic test on a sample at a stress level, S, the sample will be failed after N cycle. If the test repeated at higher stress level the cyclic to failure will be smaller. A typical plot of S-N curve in a rotating bending test of an aluminum alloy in logarithm scale is shown in **Figure 4**.

S-N data in Log–Log scale is usually in form of a straight line. An equation can be fitted on these data's is [1]:

$$
\sigma\_a = A N\_f^B \tag{5}
$$

It should be noted that Eq. 5 describes the linear part of the S-N curve and is known Baskin equation. The nonlinear part is called fatigue limit or endurance limit, Se which is seen in S-N curve of some materials like plain carbon and low alloy steels.

**Figure 4.** *S-N curve for specimens of steel. Fatigue limit can be seen at about Se = 414 MPa [30].*

This is a stress level that fatigue failure does not occur under ordinary conditions or the cycle number to failure is unlimited. It should be noted in practical applications irregular loads versus time histories are more commonly encountered [1]. Examples for these conditions are given in **Figure 5**.

For such situations that the amplitude of loading is variable, the Palmgren-Miner rule predicts fatigue life of the component [2]:

$$\frac{n\mathbf{1}}{N\mathbf{1}} + \frac{n\mathbf{2}}{N\mathbf{1}} + \frac{n\mathbf{3}}{N\mathbf{3}} + \dots + \frac{nk}{Nk} = \mathbf{1} \tag{6}$$

Where n1, n2, n3… and nk are the number of work cycles at each of the different stress levels and N2, N2, N3… and Nk are the life of the part at each similar stress levels.

According to this equation, the total life of the part is estimated from the sum of the percentage of lives spent by each stress cycle.

#### **2.2 Fracture mechanic**

The presence of cracks significantly reduces the strength and longevity of a component due to increasing the probability of occurring brittle fracture [32–34]. Cracks may be produced during the manufacturing process or other inherent flaws that convert to crack and grow until its rich critical sizes for brittle fracture. Paris equation describes the crack growth behavior of a material under cyclic loadings [1]:

$$\frac{da}{dN} = C(\Delta K)^m \tag{7}$$

$$
\Delta K = F \Delta \sigma \sqrt{\pi a} \tag{8}
$$

$$
\Delta \sigma = \sigma\_{\text{max}} - \sigma\_{\text{min}} \tag{9}
$$

Where *da dN* cyclic crack growth rate, ΔK stress intensity range, C and m are constants.

In fact this equation is log–log plot of *da dN* versus ΔK as shown in **Figure 6**. In this diagram there is a vertical part denoted Δ*Kth*, which named the fatigue crack growth threshold. This quantity is a lower limiting value that below of that the crack growth does not occur. At high growth rate the curve again become steep due to rapid unstable crack growth [1].

**Figure 5.** *Loads for one flight of a fixed-wing aircraft (a), and a simplified form of this loading [31].*

#### **Figure 6.**

*Fatigue crack growth rate for a ductile pressure vessel steel. (a) Threshold intensity factor,* Δ*Kth, (b) intermediate region which shows with power equation, (c) rapid crack growth [35].*

## **3. Fatigue damage mechanism**

Fatigue is a damage processes of components caused by cyclic loads. The process involves four stages [1, 36]:

1. Crack nucleation 2. Short crack growth 3. Long crack growth 4. Final separation

The first step is crack nucleation. It has been observed that crack of fatigue damage starts at near high stress concentration sites such as slip bands, inclusions, porosities or other manufacturing discontinuity. Localized shear plans that usually occurs at surface or within grain boundaries is another location for nucleation of fatigue cracks. This stage of fatigue cracking may be relieved with proper annealing treatment.

The next step in fatigue damage process is the crack growth. This stage is the deepening of the initial crack on the planes with maximum shear stress and it is often called crack growth stage I. This stage is greatly affected by microstructure

characteristic such as grain size, slip mode and stress level because the crack size is in the order of the microstructure.

Step 3 is the crack growth stage II. At this stage, the crack created in the previous stage grows in the direction perpendicular to the planes with high tensile stress. This stage is less affected by the microstructure because the plastic zone in crack tip is much larger than the material grain size.

Stage 4 is final separation. This stage is when the crack length reaches a critical value and the remaining cross-section cannot withstand the applied load.

Fatigue studies show that fatigue cracks usually start from a free surface. If these cracks start from the inside, the nucleation site of the crack is usually a carburized or similar surface layer [2].
