**2. Methodologies for wear modeling and simulation**

Many wear models have been developed throughout history aimed to describe and predict the phenomena. Those models are extensively reviewed in the literature [7–10] and can be broadly classified into two main categories, namely, (1) mechanistic models, based on material failure mechanism and (2) phenomenological models, based on contact mechanics.

The most popular and most used among them is the phenomenological Archard and Hirst model [11]:

$$V = k \text{Ps} \tag{1}$$

Fouvry highlighted the benefits of the energy-based wear model over the classically applied Archard's approach. This new model requires a unique experimental campaign due to the independence of the energy wear coefficient to the contact force and sliding amplitude. Both models predict the same wear under gross slip condition, since the tangential force follows the equation: *Q* = *P* where *μ* is the coefficient of friction. However, results differ under partial sliding condition, where *Q* < *P*. It is, therefore, concluded that Fouvry's model is supe-

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With the aim to solve wear modeling in more complex configurations and contact conditions, the FEM wear modeling has appeared as a good alternative in the last decade. Contact surface evolves progressively during wear phenomena, which generates an evolution in the stress state. This being so, the strategies for wear simulations consist of updating the geometry at

McColl et al. [13] introduced a methodology for numerical wear simulation based on the local implementation of the previously mentioned Archard model (the so-called modified

Δ*h*(*x*, *t*) = *k*<sup>1</sup> *p*(*x*, *t*)Δ*s*(*x*, *t*) (3)

As shown in **Figure 1**, the simulation consists of an iterative process where the local Archard's model is solved through contact pressures and sliding distributions obtained by the numerical simulation at discrete time increments. The wear value obtained at each increment is then used to update the worn geometry at the end of the cycle and the process is repeated till a

is the local Archard coefficient, *p* is the local con-

rior due to the independency of contact force and sliding amplitude.

discrete time increments, based on semi-empiric wear models.

tact pressure and Δ*s* is the relative slip distance increment.

where Δ*h* is the incremental wear depth, *k*<sup>1</sup>

predefined maximum sliding distance is reached.

**Figure 1.** Simplified wear simulation flowchart of the McColl's approach.

Archard's model):

where *V* represents the worn volume, *k* the wear coefficient (obtained experimentally), *s* the sliding distance and *P* the contact force. It is noteworthy that the wear coefficient depends on the contact force and sliding amplitude, requiring its determination for each test condition.

On the other hand, Fouvry et al. [12] proposed a new model based on the energy dissipated on the contact surface:

$$V = \alpha \sum\_{l=1}^{N} E\_{\text{d,l}} \tag{2}$$

where *α* is the energy wear coefficient and *E*d, *<sup>i</sup>* is the dissipated energy of the *i*th cycle. Fouvry highlighted the benefits of the energy-based wear model over the classically applied Archard's approach. This new model requires a unique experimental campaign due to the independence of the energy wear coefficient to the contact force and sliding amplitude. Both models predict the same wear under gross slip condition, since the tangential force follows the equation: *Q* = *P* where *μ* is the coefficient of friction. However, results differ under partial sliding condition, where *Q* < *P*. It is, therefore, concluded that Fouvry's model is superior due to the independency of contact force and sliding amplitude.

and the coefficient of friction as the most influential ones. Different regimes can be determined

• Stick regime: no slip occurs between the surfaces due to the accommodation of the dis-

• Partial-slip regime: the central zone of the contact interface is motionless or stuck while the outer one is sliding. In this case, there is a minimum wear, and fatigue failure occurs due to

• Gross-slip regime: the contact interface is in slip regime. The failure mainly occurs due to

Depending on the magnitude of stresses, fretting can cause catastrophic failure of mechanical components. It is noteworthy mentioning that fretting fatigue may reduce the lifetime of a

Despite the considerable progress made in the understanding of fretting fatigue over the last decades, it is still one of the modern issues for industrial machinery [6]. Accordingly, there is an increasing interest in the use of the finite element method (FEM) to analyze fretting phenomena, since it provides data which currently cannot be obtained through experimental testing or analytical solutions. This chapter presents a general background and the state of the art of numerical simulation and modeling of fretting in terms of wear, fatigue and fracture.

Many wear models have been developed throughout history aimed to describe and predict the phenomena. Those models are extensively reviewed in the literature [7–10] and can be broadly classified into two main categories, namely, (1) mechanistic models, based on material failure mechanism and (2) phenomenological models, based on contact mechanics.

The most popular and most used among them is the phenomenological Archard and Hirst

*V* = *kPs* (1)

where *V* represents the worn volume, *k* the wear coefficient (obtained experimentally), *s* the sliding distance and *P* the contact force. It is noteworthy that the wear coefficient depends on the contact force and sliding amplitude, requiring its determination for each test condition.

On the other hand, Fouvry et al. [12] proposed a new model based on the energy dissipated

*i* = 1 *N*

*E*d, *<sup>i</sup>* (2)

is the dissipated energy of the *i*th cycle.

depending on the slip amplitude, which is related to different failure types [4]:

placement by elastic deformation. The damage in the surface is very low.

component by half or even more, in comparison to plain fatigue [5].

**2. Methodologies for wear modeling and simulation**

crack incubation and growth (fretting fatigue).

wear (fretting wear).

196 Contact and Fracture Mechanics

model [11]:

on the contact surface:

*V* = *α* ∑

where *α* is the energy wear coefficient and *E*d, *<sup>i</sup>*

With the aim to solve wear modeling in more complex configurations and contact conditions, the FEM wear modeling has appeared as a good alternative in the last decade. Contact surface evolves progressively during wear phenomena, which generates an evolution in the stress state. This being so, the strategies for wear simulations consist of updating the geometry at discrete time increments, based on semi-empiric wear models.

McColl et al. [13] introduced a methodology for numerical wear simulation based on the local implementation of the previously mentioned Archard model (the so-called modified Archard's model):

$$
\Delta h(\mathbf{x}, t) = k\_1 p(\mathbf{x}, t) \Delta \mathbf{s}(\mathbf{x}, t) \tag{3}
$$

where Δ*h* is the incremental wear depth, *k*<sup>1</sup> is the local Archard coefficient, *p* is the local contact pressure and Δ*s* is the relative slip distance increment.

As shown in **Figure 1**, the simulation consists of an iterative process where the local Archard's model is solved through contact pressures and sliding distributions obtained by the numerical simulation at discrete time increments. The wear value obtained at each increment is then used to update the worn geometry at the end of the cycle and the process is repeated till a predefined maximum sliding distance is reached.

**Figure 1.** Simplified wear simulation flowchart of the McColl's approach.

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 of *N* actual cycles [13].

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 contact pressure data to the other part by interpolation techniques.

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

$$
\Delta h(\mathbf{x}, t) = \operatorname{aq}(\mathbf{x}, t) \Delta \mathbf{s}(\mathbf{x}, t) \tag{4}
$$

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

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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,

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).

study of the evolution of stresses and strains along the loading cycle.

which generally are applicable to both LCF and HCF regime.

As mentioned earlier, Fouvry's model shows the independency of contact force and sliding amplitude being more versatile than the commonly used Archard's equation.
