**3. Fatigue analysis approaches**

Material fatigue refers to a progressive degradation of a material caused by loading and unloading cycles. The stress fluctuations suffered over time weakens or breaks the material even at stresses lower than the yielding value. Accordingly, a lot of effort has been directed at developing fatigue-life prediction models.

Fatigue is characterized with a high scatter of the lifetime. Probabilistic approaches are recently arising in the literature to address this problem [20, 21]. However, the majority of the models currently used analyze fatigue in a deterministic way, that is, a structure fails if a given parameter reaches a critical value.

Nowadays, a variety of different approaches for fatigue life prediction exist, such as approaches based on multiaxial fatigue criteria, damage mechanics or micromechanics, which are extensively reviewed in literature [22–33]. The present review focuses on the most widely used classical methodologies, that is, multiaxial fatigue criteria.

Multiaxial fatigue criteria reduce the multiaxial stresses (usually computed by FEM analysis) to an equivalent uniaxial stress state. This way, the results can be compared to an experimental fitting curve obtained from uniaxial fatigue data. A crucial step when selecting a multiaxial criterion is to check whether the simplification from multiaxial stress state to an equivalent uniaxial stress state is acceptable or valid. This task is not simple and requires the detailed study of the evolution of stresses and strains along the loading cycle.

Since contact problems are nonlinear, the computational demand is considerable, especially in three spatial dimensions where this problem is highly magnified. Among the strategies to optimize wear simulations, the use of the cycle jumping technique should be highlighted. This approach allows to speed up the wear simulation under the assumption that wear remains constant for a small number of cycles. Therefore, a cycle jumping factor *N* multiplying the incremental wear allows using one computational wear cycle to model the material removal

A further improvement on the computational time was presented by Madge et al. [14] who programed the spatial adjustment of the contact nodes through the user defined subroutine UMESHMOTION (available on the commercial FE code Abaqus FEA). This subroutine works in an adaptive meshing constrain framework in order to adapt the mesh to the evolving geometries. Among the several benefits of using UMESHMOUTION subroutine, it should be highlighted that the updating is done incrementally through the fretting cycle, providing more stable results comparing to the updates done at the end of the cycle. Larger cycle jumps can therefore be used, decreasing significantly the computation time. However, the subroutine gives access to the pressure data of only one of the bodies, avoiding the possibility to compute wear on both parts. Cruzado et al. [15–17] overcame this limitation by transferring the available

It should be highlighted that recent publications [18, 19] proposed the use of the energy wear approach instead of the Archard's local equation. Following the previously explained frame-

Δ*h*(*x*, *t*) = *q*(*x*, *t*)Δ*s*(*x*, *t*) (4)

As mentioned earlier, Fouvry's model shows the independency of contact force and sliding

Material fatigue refers to a progressive degradation of a material caused by loading and unloading cycles. The stress fluctuations suffered over time weakens or breaks the material even at stresses lower than the yielding value. Accordingly, a lot of effort has been directed at

Fatigue is characterized with a high scatter of the lifetime. Probabilistic approaches are recently arising in the literature to address this problem [20, 21]. However, the majority of the models currently used analyze fatigue in a deterministic way, that is, a structure fails if a

Nowadays, a variety of different approaches for fatigue life prediction exist, such as approaches based on multiaxial fatigue criteria, damage mechanics or micromechanics, which are extensively reviewed in literature [22–33]. The present review focuses on the most widely used classical

contact pressure data to the other part by interpolation techniques.

work proposed by Madge, the energy equation is computed locally as:

**3. Fatigue analysis approaches**

developing fatigue-life prediction models.

given parameter reaches a critical value.

methodologies, that is, multiaxial fatigue criteria.

amplitude being more versatile than the commonly used Archard's equation.

of *N* actual cycles [13].

198 Contact and Fracture Mechanics

The book published by Socie and Marquis [22] presents a wide and detailed study about the principal multiaxial parameters, also known as Fatigue Indicator Parameters (FIPs). Those parameters can be broadly classified into three groups: strain-based, stress-based and energybased FIPs. Strain-based FIPs [34, 35] are generally related with Low-Cycle Fatigue (LCF) where plastic deformation may be predominant. Stress-based FIPs [36, 37] are associated with High-Cycle Fatigue (HCF), where the stresses usually remain in the elastic domain. Finally, energy-based models [38–40] relate the product of stresses and strains to quantify fatigue life, which generally are applicable to both LCF and HCF regime.

Additionally, fatigue can be categorized into proportional (fixed principal directions along a loading cycle) and nonproportional loading (rotation of the principal directions along a loading cycle). **Figure 2** shows the evolution of the stresses at the contact surface along fretting

**Figure 2.** Nonproportional stresses in fretting fatigue during a loading cycle: distribution of the principal stress components (*σ*11, *σ*12, and *σ*22) along the contact interface for different loading time steps (dotted red line, dashed green line, solid blue line).

fatigue cycle at three different stages of the loading cycle (maximum, mean and minimum). It can be observed that the normal stress *σ*11 and the shear stress *σ*12 fluctuate along the cycle while the stress *σ*22 remains unaltered. Consequently, principal directions rotate along the cycle generating nonproportional stresses under proportional remote loading. Therefore, the equivalent stress and strain approaches such as Von Mises criterion developed for proportional loading are not applicable in fretting since the problem is highly nonlinear and nonproportional.

For these complex stresses or loading states, other approaches such as the critical plane method are more suitable [41–43]. The critical plane method has been developed from the experimental observation of nucleation and crack growth under multiaxial loading. The critical plane models include the dominant parameters that govern the type of crack initiation and propagation. An adequate model must be one that estimates correctly both fatigue life and the dominant failure plane. However, several failure modes exist, and there is not a unique parameter that suits all.

A great deal of critical plane based FIPs have been used in the literature to assess fretting fatigue life [44–49]. Nonetheless, the most popular parameters are the energetic criteria known as Fatemi-Socie (*FS*) [39] and Smith-Watson-Topper (*SWT*) [40].

The *SWT* parameter is applied in those materials where the crack growth occurs in mode I. The critical plane is defined as the one where the product of maximum normal stress (*σ*n, max) and normal strain amplitude (*ε*n, <sup>a</sup> ) is maximum.

$$\text{SWT} = \left(\sigma\_{\text{n,max}} \varepsilon\_{\text{n,a}}\right)\_{\text{max}} \tag{5}$$

**4. Fracture modeling and simulation**

*σ*(*r*, *θ*) = ∑

singularity when *r* tends toward zero.

which may be divided into three regimes:

**Figure 3.** Singular stress field around the crack-tip.

the LEFM approach.

where *K*<sup>I</sup>

not propagate.

Fracture mechanics is the field of mechanics concerned with the study of structures integrity in the presence of cracks. Within this field, there are several approaches, such as the linear elastic fracture mechanics (LEFM), the non linear fracture mechanics (NLFM) or the elastoplastic fracture mechanics (EPFM) [53–55]. This chapter focuses on the most widely used one,

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

From a fully elastic point of view, Williams [56] presented an eigen function expansion method that provides a framework for the description of the stress state near a crack-tip. For each cracked configuration, a sequence of coefficients depending on the geometry and load describes

> *i Ki* \_1 √ \_ <sup>2</sup>*<sup>r</sup> f i*

assumes that the singular stresses dominate the stress field near the crack front, thus neglecting higher order terms of the Williams series. It can be easily seen that the stress field shows a

As far as fatigue crack growth behavior is concerned, this is usually described by the relationship between the crack growth length increase per cycle (d*a*/d*N*) as a function of the SIF range (Δ*K*). The typical log-log plot of crack growth behavior is shown schematically in **Figure 4**,

• Regime I: the near threshold region—below the threshold value of SIF (Δ *K*th) cracks will

is the Stress Intensity Factor (SIF) for each of the fracture mode. The use of SIFs

(*θ*) (7)

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the stress state with respect to the radius of circumference *r* and angle *θ* (see **Figure 3**).

Irwin identified the first physically valid term in this infinite series, the *K* field [57]:

Under shear loading condition, crack lip surfaces generate frictional forces that reduce stresses at the crack tip, thus increasing fatigue life. However, tensile stresses and strains will separate the crack surfaces, reducing the friction forces. The energetic FIP *FS* can be understood as the cyclic shear strain to include the crack closure effect multiplied by normal stress to take into account the opening of the crack. *FS* <sup>=</sup> <sup>Δ</sup> *<sup>γ</sup>* \_max

$$FS = \frac{\Lambda^{\gamma}}{2} \{ 1 + k\_{\rm FS}^{\prime} \frac{\sigma\_{\rm \gamma \rm res}}{\sigma\_{\rm \gamma}} \} \tag{6}$$

where Δ *γ*max is the maximum range of shear strain on any plane, *σ*n, max is the maximum normal stress in that particular plane, *σ*<sup>y</sup> is the material yield stress, *k*FS is a material dependent factor.

Vázquez et al. [50] recently compared both parameters for the analysis of the initial crack path in cylindrical fretting contact, concluding that the *SWT* parameter gives much better correlation than the *FS* parameter.

It should be mentioned that these parameters give a local life prediction and seek to find the hot spot to give the minimum life estimation. However, when high stress gradient events appear, for example, fretting case, an over-estimation of crack nucleation is predicted at the hot spot. Consequently, a nonlocal approach such as the Theory of Critical Distances (TCD) [51] used extensively in notched fatigue is recommended [52].
