**3. Statistical analysis**

The loads acquired are compared with S-N curves. The S-N curves are often expressed on semi-log, normal and log–log coordinates. **Figure 5** shows a schematic curve on different coordinates, linear, semi-log and log–log. The most common representation is log-log since it becomes linear (**Figure 5c**).

The S-N curve represents the material or component fatigue strength, and is split into regions depending on its cycles. Extremely low cycle fatigue (ELCF) is defined from 0 until 100 cycles, between this limit and until 1000 cycles is low cycle fatigue (LCF), and between 1000 cycles and until 1 × 106 for steel and 5 × 107 for nodular cast iron is defined as high cycle fatigue (HCF). Anything beyond this point is defined as very high cycle fatigue (VHCF) [7].

To compare the S-N curve with loads, the time history is analyzed. **Figure 6** shows a schematic waveform. The main characteristic is the stress amplitude S<sup>a</sup> . If it has constant amplitude, the stress range SR is constant and is defined by the difference of the maximum stress (Smax) and minimum (Smin) in a cycle (Eqs. (1)–(3)).

$$\mathcal{S}\_{\mathbb{R}} = \mathcal{S}\_{\text{max}} - \mathcal{S}\_{\text{min}} \tag{1}$$

$$S\_a = \frac{S\_k}{2} = \frac{S\_{mr} - S\_{min}}{2} \tag{2}$$

The mean stress Sm is defined as

$$\mathcal{S}\_m = \frac{\mathcal{S}\_{\text{max}} - \mathcal{S}\_{\text{min}}}{2} \tag{3}$$

A fully alternating stress S<sup>m</sup> = S<sup>a</sup> .

The stress ratio *(R)* is defined as the ratio of minimum to maximum stress as is shown in Eq. (4).

$$R = \frac{S\_{\min}}{S\_{\max}} \tag{4}$$

cycle counting methods such as the Rainflow used to extract cycles from random histories in the time domain [13, 14], based on the analogy of raindrops falling on a roof. **Figure 8** shows

Accelerated Fatigue Test in Mechanical Components http://dx.doi.org/10.5772/intechopen.72640 259

The cycle counting is represented in a matrix based on **Figure 5**. The signal has 2 cycles from 5 to 3, 1 cycle from 6 to 3, 1 cycle from 1 to 5, 1 cycle from 2 to 4 (**Figure 5a**), 2 cycles from 1 to 6 (**Figure 5b**), and it has residue. In **Figure 5c**, these cycles are tabulated on a matrix, which

It is possible to evaluate time histories with other types of cycle counting methods, such as the level crossing method [15] where the amplitudes of the loads are split into a number of levels based on ranges, and the load is counted when it has peak at a different level, changing its slope from positive to negative or negative to positive; the cycle counting is shown in **Figure 9**. In the range pair counting method, the magnitude of loads is split into a number of levels. The result of the extracted number of reversals is shown tabulated in **Figure 10b**. **Table 2** sum-

Rainflow counting process.

**Figure 7.** Stress ratio.

**Figure 6.** Signal characteristics.

depending on its counting can be represented by colors.

marizes the events counted in **Figure 10**.

The fatigue damage of a component is influenced in high cycle region by the mean stress expressed by its stress ratio. In normal *R* ≥ 0, open microcracks accelerate the propagation of stress, while *R* = ∞ or >1 closes the microcrack that is beneficial for fatigue strength. In low cycle fatigue region, the plastic deformation eliminates the effect of mean stress to improve or detriment the fatigue strength. The schematic stress ratio is shown in **Figure 7**.

The amplitude ratio is the ratio of the stress amplitude to mean stress as show in Eq. (5).

$$A = \frac{S\_s}{S\_u} = \frac{1 - R}{1 + R} \tag{5}$$

The loads are monitored with cycle counting that is used to summarize variable amplitude time histories, providing the repetitions of the load during the time history. There are different

**Figure 6.** Signal characteristics.

**3. Statistical analysis**

258 Contact and Fracture Mechanics

becomes linear (**Figure 5c**).

The loads acquired are compared with S-N curves. The S-N curves are often expressed on semi-log, normal and log–log coordinates. **Figure 5** shows a schematic curve on different coordinates, linear, semi-log and log–log. The most common representation is log-log since it

The S-N curve represents the material or component fatigue strength, and is split into regions depending on its cycles. Extremely low cycle fatigue (ELCF) is defined from 0 until 100 cycles, between this limit and until 1000 cycles is low cycle fatigue (LCF), and between 1000 cycles and until 1 × 106 for steel and 5 × 107 for nodular cast iron is defined as high cycle fatigue

To compare the S-N curve with loads, the time history is analyzed. **Figure 6** shows a sche-

tude, the stress range SR is constant and is defined by the difference of the maximum stress

*SR* = *Smax* − *Smin* (1)

The stress ratio *(R)* is defined as the ratio of minimum to maximum stress as is shown in

The fatigue damage of a component is influenced in high cycle region by the mean stress expressed by its stress ratio. In normal *R* ≥ 0, open microcracks accelerate the propagation of stress, while *R* = ∞ or >1 closes the microcrack that is beneficial for fatigue strength. In low cycle fatigue region, the plastic deformation eliminates the effect of mean stress to improve or detriment the fatigue strength. The schematic stress ratio is

The amplitude ratio is the ratio of the stress amplitude to mean stress as show in Eq. (5).

\_\_\_*a Sm*

The loads are monitored with cycle counting that is used to summarize variable amplitude time histories, providing the repetitions of the load during the time history. There are different

= \_\_\_\_ <sup>1</sup> <sup>−</sup> *<sup>R</sup>*

\_\_\_\_ *min Smax*

<sup>2</sup> <sup>=</sup> *Smax* <sup>−</sup> *<sup>S</sup>* \_\_\_\_\_\_\_\_*min*

. If it has constant ampli-

(4)

<sup>2</sup> (2)

<sup>2</sup> (3)

<sup>1</sup> <sup>+</sup> *<sup>R</sup>* (5)

(HCF). Anything beyond this point is defined as very high cycle fatigue (VHCF) [7].

matic waveform. The main characteristic is the stress amplitude S<sup>a</sup>

(Smax) and minimum (Smin) in a cycle (Eqs. (1)–(3)).

*Sm* <sup>=</sup> *Smax* <sup>−</sup> *<sup>S</sup>* \_\_\_\_\_\_\_\_*min*

*<sup>R</sup>* <sup>=</sup> *<sup>S</sup>*

*<sup>A</sup>* <sup>=</sup> *<sup>S</sup>*

.

*Sa* <sup>=</sup> *<sup>S</sup>*\_\_*<sup>R</sup>*

The mean stress Sm is defined as

A fully alternating stress S<sup>m</sup> = S<sup>a</sup>

Eq. (4).

shown in **Figure 7**.

cycle counting methods such as the Rainflow used to extract cycles from random histories in the time domain [13, 14], based on the analogy of raindrops falling on a roof. **Figure 8** shows Rainflow counting process.

The cycle counting is represented in a matrix based on **Figure 5**. The signal has 2 cycles from 5 to 3, 1 cycle from 6 to 3, 1 cycle from 1 to 5, 1 cycle from 2 to 4 (**Figure 5a**), 2 cycles from 1 to 6 (**Figure 5b**), and it has residue. In **Figure 5c**, these cycles are tabulated on a matrix, which depending on its counting can be represented by colors.

It is possible to evaluate time histories with other types of cycle counting methods, such as the level crossing method [15] where the amplitudes of the loads are split into a number of levels based on ranges, and the load is counted when it has peak at a different level, changing its slope from positive to negative or negative to positive; the cycle counting is shown in **Figure 9**.

In the range pair counting method, the magnitude of loads is split into a number of levels. The result of the extracted number of reversals is shown tabulated in **Figure 10b**. **Table 2** summarizes the events counted in **Figure 10**.

**Figure 8.** Rainflow counting process: (a) initial counting, (b) continue counting, and (c) residue.

The signals can be seen in time domain and frequency domain. A transfer function can be used in the frequency domain to relate the power spectral density (PSD) of the input desired load to the PSD of the output stress (Eq. (6)) [16]:

$$
\sigma\_{\rm pSD}(w) = \left| |h(w)|^2 F\_{\rm pSD}(w) \right. \tag{6}
$$

damage the components by itself, the accumulated damage induced by high loads can be propagated by such small loads. The correction factors for the slope depend on the material [18].

Accelerated Fatigue Test in Mechanical Components http://dx.doi.org/10.5772/intechopen.72640 261

**Figure 9.** Level crossing counting process: (a) time history, (b) quantity per class, (c) histogram, and (d) absolute

**Figure 10.** Range pair counting process, (a) time signal and (b) events.

cumulative frequency.

here the squaring process is required to get the transfer function in the correct units of PSD stress [17]. In this equation, σPSD (w) is the PSD of the stress at frequency *w* (N2 /Hz); *h* (W) is the linear transfer function at frequency w; and FPSD (w) is the PSD of the input amplitude at frequency w (N2 /Hz). The advantage of analyzing the responses with PSD is that it helps us represent the energy of the time signal at each frequency.

The time histories for constant amplitude test spectrum is linear (**Figure 11a**), and for variable amplitude, it is a curve (**Figure 11b**) generated by the cycle counting. Although a theoretical fatigue limit has been proposed, with the introduction of new test equipment for high frequency, the prediction has been improved at low load levels. Although this stress couldn't damage the components by itself, the accumulated damage induced by high loads can be propagated by such small loads. The correction factors for the slope depend on the material [18].

**Figure 9.** Level crossing counting process: (a) time history, (b) quantity per class, (c) histogram, and (d) absolute cumulative frequency.

**Figure 10.** Range pair counting process, (a) time signal and (b) events.

The signals can be seen in time domain and frequency domain. A transfer function can be used in the frequency domain to relate the power spectral density (PSD) of the input desired

*σPSD*(*w*) = |*h*(*w*)|2 *FPSD*(*w*) (6)

here the squaring process is required to get the transfer function in the correct units of PSD

the linear transfer function at frequency w; and FPSD (w) is the PSD of the input amplitude at

The time histories for constant amplitude test spectrum is linear (**Figure 11a**), and for variable amplitude, it is a curve (**Figure 11b**) generated by the cycle counting. Although a theoretical fatigue limit has been proposed, with the introduction of new test equipment for high frequency, the prediction has been improved at low load levels. Although this stress couldn't

/Hz). The advantage of analyzing the responses with PSD is that it helps us

/Hz); *h* (W) is

stress [17]. In this equation, σPSD (w) is the PSD of the stress at frequency *w* (N2

**Figure 8.** Rainflow counting process: (a) initial counting, (b) continue counting, and (c) residue.

load to the PSD of the output stress (Eq. (6)) [16]:

represent the energy of the time signal at each frequency.

frequency w (N2

260 Contact and Fracture Mechanics


**Table 2.** Range pair counting.

loads from the users and different responses are taken into account. *Ca*

is the average driver, and *Cc*

or below 1 [4]. The failure is observed when there is a physical crack.

*<sup>D</sup>* <sup>=</sup> <sup>∑</sup> *<sup>n</sup>*

and *Cd*

components but for more time. After having extrapolated the goal use, the spectrum is built considering all these measurements. **Figure 13** shows a schematic spectrum development. For variable amplitude test, the spectrum is reproduced and monitored with statistical analysis.

The repetitions of the cycles are found using the linear damage rule of Miner (Eq. (7)), and damage is evaluated using the ratio of the loads *(n)* with the number of repetitions (*N)* toler-

The damage could be reached when the summation is 1 and there is an effect of sequence load. Depending on sequence effect loads, the damage can be reached too with values above

**Figure 2** shows that the first point to acquire information is to install measurement devices on the component or in its vicinity, to obtain the responses of the components that induce

\_\_*i Ni*

aggressive driver, *Cb*

**Figure 13.** Schematic spectrum development.

**Figure 12.** Spectrum development target.

ated at *i* load level.

**5. Instrumentation**

is the driver A, the most

(7)

are the drivers that use less aggressive

Accelerated Fatigue Test in Mechanical Components http://dx.doi.org/10.5772/intechopen.72640 263

**Figure 11.** Schematic spectrums versus components S-N curves: (a) constant amplitude, (b) variable amplitude.
