**5. Fretting fatigue numerical simulation**

Over the years, a significant number of different methodologies for fretting fatigue life prediction have been presented. First models were purely analytical, and numerical models appeared as computers evolved. Nowadays the preferred approaches can be classified into hybrid (combining analytical and numerical methods) or fully numerical methods.

Fretting fatigue life estimation has become an object of interest in the literature of the field. In general, the study of fretting is divided into two stages, the crack initiation (*N*<sup>i</sup> ) and its subsequent propagation (*N*p). Therefore, the sum of the two stages gives a total life prediction:

$$N\_t = N\_i + N\_p \tag{10}$$

separately for a set of points at different depths, considering the distance from the surface to the point under analysis as the initial crack length. Fretting fatigue life estimation is then calculated for each point (*N*<sup>i</sup> + *N*p) and the minimum of those predictions is considered the estimated life. This methodology combines the FEM for crack initiation analysis and analytical models for crack propagation phase. It has been shown that the use of the X-FEM for crack propagation analysis following the same framework gives a better correlation in fatigue life prediction [47]. More recently, the use of nonlocal methods such as the TCD (the reader is referred to Section 3) was suggested by some authors in order to define the location of the multiaxial analysis and

**Figure 5.** Dependency of fatigue analysis location and initial crack length in fretting fatigue lifetime. (a) Von Mises stress contour showing the high stress gradient present in fatigue phenomena. (b) and (c) Schematic representation of the

Fretting: Review on the Numerical Simulation and Modeling of Wear, Fatigue and Fracture

) and the influence of initial crack length

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influence of fatigue analysis location in the crack initiation cycle prediction (*N*<sup>i</sup>

Despite fretting wear and fretting fatigue being frequently linked, most of the approaches, such as the ones mentioned earlier, neglect wear phenomena. A further improvement is therefore to consider the wear simulation in order to study the effect of wear on fatigue life.

In this regard, one of the most prominent work was presented by Madge et al. [84] In this work, the 2D numerical simulation of material removal process is performed first under the UMESHMOTION framework (the reader is referred to the Section 2). Then, the multiaxial fatigue analysis coupled with a damage accumulation framework is carried out in order to account for the effect of wear on fatigue lifetime. Finally, the propagation phase is analyzed via submodeling technique, which allows to transfer the stress state of the contact surface from global wear model to crack submodel. It should be noted that this approach does not account for the explicit interaction between the fretting contact and the crack. This study showed that the linkage of wear modeling with fatigue analysis is a key factor to successfully

A further improvement of the previous approach was recently published by the present authors [85] by combining wear, fatigue and fracture phenomena in a single numerical model.

**Figure 6** shows the flow chart of the coupled numerical analysis approach.

to set the initial crack length [49, 83].

in the propagation cycle estimation (*N*p), respectively.

predict the increase in life in the gross sliding regime.

In those situations in which the initiation phase dominates over propagation, some authors propose to estimate life considering only the initiation stage [77]. Conversely, for cases where crack growth stage dominates the components' life, the initiation phase is sometimes neglected [78]. Nonetheless, most current approaches combine both phases in order to provide a total life prediction, and this chapter is therefore centered on those combined models.

The number of cycles to initiate a crack is typically obtained with a FIP (the reader is referred to Section 3 for multiaxial criterion details). Crack propagation is subsequently considered using different fatigue crack growth laws, which require the definition of an initial crack length.

It is noteworthy that the development of a combined initiation-propagation model involves two critical steps: (1) the selection of the stress-deformation analysis location and (2) the definition of the initial crack length. On the one hand, due to the high stress gradient that characterizes fretting (see **Figure 5(a)**), the predicted crack nucleation cycles will increase as the location is further from the contact (see **Figure 5(b)**. On the other hand, longer cracks grow faster, leading to a lower number of propagation cycles (see **Figure 5(c)**).

Among the different methodologies presented for fretting fatigue lifetime prediction, the simplest approach is to perform the multiaxial analysis at the hot spot (i.e., at the surface), which results in a conservative crack initiation prediction [79]. Once the condition of crack initiation is reached, a predefined crack length is introduced in the numerical model [44, 79]. Alternatively, other authors perform the multiaxial analysis at a certain distance from the surface instead of at the hot spot, obtaining less conservative life estimations [14, 78, 80, 81]. However, the selected distance is usually arbitrary and there is no consensus in the literature.

A further improvement was introduced by Navarro et al. [82] who presented a nonarbitrary criteria. In this methodology, both the initiation and propagation phases are computed Fretting: Review on the Numerical Simulation and Modeling of Wear, Fatigue and Fracture http://dx.doi.org/10.5772/intechopen.72675 205

of each of the method depends on whether the problem is two or three-dimensional. Although the benefits of the LSM or FMM are not overwhelming in 2D, the use of implicit methods becomes necessary in 3D since the explicit representation can be quite difficult to discretize

Over the years, a significant number of different methodologies for fretting fatigue life prediction have been presented. First models were purely analytical, and numerical models appeared as computers evolved. Nowadays the preferred approaches can be classified into

Fretting fatigue life estimation has become an object of interest in the literature of the field. In

sequent propagation (*N*p). Therefore, the sum of the two stages gives a total life prediction:

*N*<sup>f</sup> = *N*<sup>i</sup> + *N*<sup>p</sup> (10)

In those situations in which the initiation phase dominates over propagation, some authors propose to estimate life considering only the initiation stage [77]. Conversely, for cases where crack growth stage dominates the components' life, the initiation phase is sometimes neglected [78]. Nonetheless, most current approaches combine both phases in order to provide a total life prediction, and this chapter is therefore centered on those combined models. The number of cycles to initiate a crack is typically obtained with a FIP (the reader is referred to Section 3 for multiaxial criterion details). Crack propagation is subsequently considered using different fatigue crack growth laws, which require the definition of an initial crack length.

It is noteworthy that the development of a combined initiation-propagation model involves two critical steps: (1) the selection of the stress-deformation analysis location and (2) the definition of the initial crack length. On the one hand, due to the high stress gradient that characterizes fretting (see **Figure 5(a)**), the predicted crack nucleation cycles will increase as the location is further from the contact (see **Figure 5(b)**. On the other hand, longer cracks grow

Among the different methodologies presented for fretting fatigue lifetime prediction, the simplest approach is to perform the multiaxial analysis at the hot spot (i.e., at the surface), which results in a conservative crack initiation prediction [79]. Once the condition of crack initiation is reached, a predefined crack length is introduced in the numerical model [44, 79]. Alternatively, other authors perform the multiaxial analysis at a certain distance from the surface instead of at the hot spot, obtaining less conservative life estimations [14, 78, 80, 81]. However, the selected distance is usually arbitrary and there is no consensus in the literature. A further improvement was introduced by Navarro et al. [82] who presented a nonarbitrary criteria. In this methodology, both the initiation and propagation phases are computed

faster, leading to a lower number of propagation cycles (see **Figure 5(c)**).

) and its sub-

hybrid (combining analytical and numerical methods) or fully numerical methods.

general, the study of fretting is divided into two stages, the crack initiation (*N*<sup>i</sup>

[75, 76].

204 Contact and Fracture Mechanics

**5. Fretting fatigue numerical simulation**

**Figure 5.** Dependency of fatigue analysis location and initial crack length in fretting fatigue lifetime. (a) Von Mises stress contour showing the high stress gradient present in fatigue phenomena. (b) and (c) Schematic representation of the influence of fatigue analysis location in the crack initiation cycle prediction (*N*<sup>i</sup> ) and the influence of initial crack length in the propagation cycle estimation (*N*p), respectively.

separately for a set of points at different depths, considering the distance from the surface to the point under analysis as the initial crack length. Fretting fatigue life estimation is then calculated for each point (*N*<sup>i</sup> + *N*p) and the minimum of those predictions is considered the estimated life. This methodology combines the FEM for crack initiation analysis and analytical models for crack propagation phase. It has been shown that the use of the X-FEM for crack propagation analysis following the same framework gives a better correlation in fatigue life prediction [47].

More recently, the use of nonlocal methods such as the TCD (the reader is referred to Section 3) was suggested by some authors in order to define the location of the multiaxial analysis and to set the initial crack length [49, 83].

Despite fretting wear and fretting fatigue being frequently linked, most of the approaches, such as the ones mentioned earlier, neglect wear phenomena. A further improvement is therefore to consider the wear simulation in order to study the effect of wear on fatigue life.

In this regard, one of the most prominent work was presented by Madge et al. [84] In this work, the 2D numerical simulation of material removal process is performed first under the UMESHMOTION framework (the reader is referred to the Section 2). Then, the multiaxial fatigue analysis coupled with a damage accumulation framework is carried out in order to account for the effect of wear on fatigue lifetime. Finally, the propagation phase is analyzed via submodeling technique, which allows to transfer the stress state of the contact surface from global wear model to crack submodel. It should be noted that this approach does not account for the explicit interaction between the fretting contact and the crack. This study showed that the linkage of wear modeling with fatigue analysis is a key factor to successfully predict the increase in life in the gross sliding regime.

A further improvement of the previous approach was recently published by the present authors [85] by combining wear, fatigue and fracture phenomena in a single numerical model. **Figure 6** shows the flow chart of the coupled numerical analysis approach.

study of the fretting phenomena as a whole, allowing to model explicitly the interaction between the fretting contact and the crack. It is expected that further analysis using this framework will allow a better understanding of the synergies between wear, fatigue and fracture phenomena.

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The complex nature of the physical phenomenon of fretting involves the mixture of different fields of engineering such as fatigue, fracture and tribology. This chapter has introduced the state of the art of the currently available modeling and simulation methods to analyze fretting phenomenon. The benefits and drawbacks of each reviewed technique have been highlighted. Finally, a numerical architecture of coupled wear, fatigue and fracture methodology has been

Among the main open challenges identified, a predominant role is taken by the need to compare the increasing amount of methodologies to assess fretting lifetime, and it is likely to become the subject of considerable debate in the research community in the near future.

, Alaitz Zabala<sup>1</sup>

2 Structural Mechanics and Design, Engineering Faculty, Mondragon Unibertsitatea,

[2] Collins JA, Marco SM. The effect of stress direction during fretting on subsequent fatigue

[3] Dobromirski JM. Variables in the Fretting Process: Are There 50 of Them? Philadelphia,

[5] Hills DA, Nowell D. Mechanics of Fretting Fatigue. 1st ed. Springer Nederlands: Kluwer

, Miren Larrañaga<sup>2</sup>

and Xabier Gomez<sup>1</sup>

introduced, which allows to analyze the fretting phenomenon as a whole.

1 Surface Technologies, Engineering Faculty, Mondragon Unibertsitatea,

[4] Vingsbo O, Söderberg S. On fretting maps. Wear. 1988;**126**:131-147

**6. Summary and concluding remarks**

**Author details**

Iñigo Llavori1,2\*, Jon Ander Esnaola<sup>2</sup>

Arrasate-Mondragon, Spain

Arrasate-Mondragon, Spain

PA: ASTM; 1992

Academic Publisher; 1994

**References**

\*Address all correspondence to: illavori@mondragon.edu

[1] Waterhouse RB. Fretting wear. Wear. 1984;**100**:107-118

life. Proceedings of ASTM. 1964;**64**:547-560

**Figure 6.** Flowchart showing computational sequence of the coupled wear, fatigue and fracture in fretting numerical analysis developed by [85].

The simulation algorithm is divided into two blocks corresponding to initiation and propagation stages, respectively. The first stage runs under the FEM framework, where the accumulated damage is computed during wear simulation iteratively. This process is repeated until the accumulated damage value reaches the value of 1 (Miner ≥ 1), that is, up to concluding the initiation phase. Afterwards, the propagation is computed through X-FEM code published by Giner et al. [76]. In this stage, the crack propagation is calculated during wear simulation iteratively up to reaching the failure criteria (Δ*K* ≥ Δ *K*<sup>c</sup> ). The presented model allows the study of the fretting phenomena as a whole, allowing to model explicitly the interaction between the fretting contact and the crack. It is expected that further analysis using this framework will allow a better understanding of the synergies between wear, fatigue and fracture phenomena.
