**4. Fracture modeling and simulation**

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

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

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

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)

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

<sup>2</sup> (1 + *k*FS

where Δ *γ*max is the maximum range of shear strain on any plane, *σ*n, max is the maximum nor-

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 correla-

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)

*σ*\_n, max

*<sup>σ</sup>*<sup>y</sup> ) (6)

is the material yield stress, *k*FS is a material dependent

(5)

) is maximum.

known as Fatemi-Socie (*FS*) [39] and Smith-Watson-Topper (*SWT*) [40].

*SWT* <sup>=</sup> (*σ*n, max *<sup>ε</sup>*n, <sup>a</sup>)max

[51] used extensively in notched fatigue is recommended [52].

nonproportional.

200 Contact and Fracture Mechanics

parameter that suits all.

and normal strain amplitude (*ε*n, <sup>a</sup>

account the opening of the crack.

mal stress in that particular plane, *σ*<sup>y</sup>

tion than the *FS* parameter.

factor.

*FS* <sup>=</sup> <sup>Δ</sup> *<sup>γ</sup>* \_max

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, the LEFM approach.

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 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]:

$$
\sigma(r\_\prime, \theta) = \sum\_l K\_{\parallel} \frac{1}{\sqrt{2\pi}} f\_l(\theta) \tag{7}
$$

where *K*<sup>I</sup> is the Stress Intensity Factor (SIF) for each of the fracture mode. The use of SIFs 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 singularity when *r* tends toward zero.

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**, which may be divided into three regimes:

• Regime I: the near threshold region—below the threshold value of SIF (Δ *K*th) cracks will not propagate.

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

**4.1. Stress intensity factor computation**

the new crack boundaries.

{*FL*(**x**)} ≡ √

at the crack tip (see **Figure 3**).

**u**<sup>X</sup>‐FEM(**x**) = ∑

shape function, **u***<sup>i</sup>*

mation to the displacement field, defined as:

the degrees of freedom of the enriched nodes.

*i* ∈ *I* **u***<sup>i</sup> Ni*

capable of extracting SIFs for each mode separately (*K*<sup>I</sup>

element method (X-FEM) as a solution to overcome these issues.

reproduce the singular behavior of LEFM by the following expression:

<sup>2</sup>), cos(*<sup>θ</sup>*\_

(**x**) + ∑ *i* ∈ *L* **a***<sup>i</sup> Ni*

\_ *r*{sin(*<sup>θ</sup>*\_

Many of the initial analytical approaches initially developed for SIFs calculation [64–67] have been now outdated by the versatility offered by numerical methods. In this regard, one of the main methods is the interaction integral [68] through the equivalent domain integral [69] (nowadays implemented in commercial finite element codes such as Abaqus FEA). The interaction integral is an extension of the well-known *J* integral proposed by Rice [70], which is

The computation of the interaction integral requires first to compute the stresses and strains by means of the FEM. However, the study of the singular problem through the FEM presents several drawbacks. On the one hand, in the classical formulation of the FEM the element edges need to conform to the crack boundaries, which require the use of cumbersome meshing techniques. On the other hand, the shape functions employed are generally of low-order polynomials, leading to the use of very refined mesh to compute reliable stresses and strains around the crack tip. Additionally, fatigue crack requires remeshing techniques to conform to

Once Melenk and Babuška [71] showed that the finite elements could be enriched with additional functions to represent a given function, Möes et al. [72] proposed the eXtended finite

In the X-FEM it is not necessary to have a mesh that conforms to the crack geometry, thus the finite element mesh is independent of the crack shape. To this end, the FEM model is enriched with additional degrees of freedom. On the one hand, the Heaviside function (*H*(**x**) = ±1) is used to introduce discontinuity along the crack faces. On the other hand, the FEM model is additionally enriched with the asymptotic function derived by Irwin, and, therefore, can

<sup>2</sup>), sin(*<sup>θ</sup>*\_

where *r*, *θ* represents the polar coordinates defined with respect to the local reference system

Therefore, the classical FEM displacement definition is enriched to obtain the X-FEM approxi-

where *I* is the set of all nodes in the mesh, *L* and *K* are the sets of the enriched nodes, *Ni*

is the classical degree of freedom of the FEM and **a***<sup>i</sup>*

Another important aspect of the X-FEM is the geometrical representation of the evolving cracks and the definition of the elements and nodes to be enriched. There are several methods to perform this task and can be divided into two groups: (1) implicit methods, such as the Level Set Method (LSM) [73] or the fast version called the Fast Marching Method (FMM) and (2) explicit methods, such as geometric predicates [74] or other approaches [72]. The suitability

<sup>2</sup>) sin(*θ*), cos(*<sup>θ</sup>*\_

(**x**)*H*(**x**) + ∑

*i* ∈ *K Ni* (**x**) ( ∑ *l* = 1 4 **b***i <sup>l</sup> Fl* (**x**)

<sup>2</sup>) sin(*θ*)} (8)

 and **b***<sup>i</sup> l* ) (9)

are respectively

is the

, *K*II and *K*III).

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

http://dx.doi.org/10.5772/intechopen.72675

203

**Figure 4.** Schematic representation of a typical fatigue crack growth rate curve plotted in log-log scale.


Regarding the simulation and modeling of crack growth behavior, the computation of the SIFs have been a priority in fracture mechanics, which has given rise to a great diversity of techniques (discussed in Section 4.1). In cases where the SIF range overcomes the threshold value (Δ*K* > Δ *K*th), the velocity and direction of the crack growth should be computed.

Crack propagation rate is usually described by means of a phenomenological law of the type d*a*/d*N* = *f*(Δ*K*). A comprehensive analysis of crack propagation velocity models in fretting fatigue was carried out by Navarro et al. [58] who analyzed nine of the fatigue crack growth models for an aluminum alloy Al7075. This study concluded that the Paris law [59] and the modified SIF model [60] were the most suitable ones for the experimental campaign carried out.

Concerning crack orientation criteria, they are generally based on the analysis of the stress and strain fields. The suitability of each criterion mainly depends on the evolution of the stresses and strains along a loading cycle. It should be noted that fretting fatigue usually induces friction between crack faces prone to slip motion during the loading cycle. The problem is therefore nonproportional, and the classical orientation criteria for proportional loading such as the maximum circumferential stress [61] or the minimum of the strain energy density factor *S* [62] among others are not applicable. Giner et al. [63] analyzed the suitability of the nonproportional loading criteria available in the literature for fretting fatigue problem, concluding that the prediction of the crack path observed in the complete contact experiments did not present a good agreement with the models available. Therefore, they developed the criterion of the minimum shear stress range, which is a generalization for nonproportional loading of the so-called criterion of local symmetry well established for proportional loading. The numerical results obtained by this new criterion were in good agreement with the experimental observation.
