# On the Convergence of Stresses in Fretting Fatigue

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## Abstract

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## 1. Introduction

_{axial}. On application of the bulk stress, the compliance springs transmit an oscillatory tangential force, Q, at the pads. Generally, the tangential load |Q| is smaller than the product of the normal load, F, by the coefficient of friction µ and the contact is divided into two regions: A stick zone and a slip region. In the early 1970s, Nishioka and Hirakawa [7] had already used this configuration to study the effects of slip amplitude in the fatigue strength of specimens. Even in recent research, this test set-up is still very common. For instance, Pierres et al. [8] proposed a combined numerical and experimental approach to simulate fretting fatigue crack growth of 2D and 3D configurations. A similar methodology was used by Luke et al. [9], however, they were interested in simulating crack initiation using different damage parameters and they used laboratory tests to validated their predictions.

_{axial}, respectively). However, these solutions are valid under a series of conditions, such as infinite and idealized bodies, elastic material properties, and loading conditions, among others. In addition, the stress field near the contact region is variable, multiaxial and non-proportional [13], which provides extra complexity to the phenomena.

_{axial}on the contact stresses distributions, and compared the results with analytical approximations, validating the latter. Tur et al. [18] treated the problem considering the effects of plasticity on the contact stress distribution for a Titanium material and analyzed the impact of plastic deformations on the size of the stick zone and peak stresses. They concluded that the plastic zone started at the trailing edge (the edge of the largest slip zone) and that the effects of contact stresses decayed rapidly as the distance from the contact increased.

## 2. Analytical Solutions

#### 2.1. Hertzian Solutions for the Pressure Distribution

- Contact surface profiles are smooth, continuous and nonconforming;
- Small strains at contact region;
- Bodies can be approximated as a semi-infinite elastic half-space near the contact zone;
- Frictionless contact.

_{max}is the maximum contact pressure at the center of the contact; R is the combined curvature; and E

^{∗}is the combined modulus of elasticity. Both R and E

^{∗}can be defined as:

_{i}, for i = 1,2 are the Young’s Modulus and ν

_{i}, for i = 1,2 are the Poisson’s ratio for the first and second bodies, respectively. The flat specimen can be considered as a cylinder with an infinitely large radius R

_{1}= ∞ and the combined curvature, R, becomes equal to the radius of the surface of the pad R

_{2}.

#### 2.2. Solutions for Combined Normal and Tangential Loads

#### 2.3. Effect of Bulk Load σ_{axial} on Contact Shear Traction

_{axial}. This causes an eccentricity to the solution presented in Section 2.2, and for the case of negative tangential load, it can be written as [12]:

_{1}) and q(x

_{2}), at the leading edge and at trailing edge sides (the edge of the largest slip zone [20]), respectively.

#### 2.4. Effect of Bulk Load σ_{axial} on Subsurface Stresses

_{2}). Other research [12,21,22] has also pointed out that the principal crack initiates near the trailing edge (x = a). The reason for that may be related to the contribution of the principal stress σ

_{xx}in the stress state at the contact interface. As discussed by Szolwinski and Farris [23], studies showed that the sharp peak in tangential stresses σ

_{xx,max}, at trailing edge of the contact region (see Figure 4), might play a significant role on fretting fatigue crack initiation.

_{xx}, as function of x for a given normal and tangential loads (F and Q) and coefficient of friction, µ, in the slip zone [12,19,24]. For instance, Szolwinski and Farris [24] provided an analytical solution for the stress distribution, σ

_{xx}, treating the problem as a superposition of individual stress components, caused by the normal pressure distribution and surface tractions, q’(x) and q’’(x).

_{axial}brings some extra complexity to the problem, there are still some simplified equations to estimate stresses at contact. McVeigh and Farris [17] adjusted the analytical solution from Szolwinski and Farris [24] by adding bulk stress in the distribution of σ

_{xx}. Szolwinski and Farris [23], based on the work done by McVeigh and Farris [17], provided a simplified equation to estimate the maximum peak stress σ

_{xx,max}as:

## 3. Finite Element Model: Cylinder Pad on Flat Specimen

^{®}and an analysis of the fretting cycle was performed, aiming to study the model response to different mesh sizes. Three values of coefficients of friction were considered (0.3, 0.85 and 2.0). These variable values of coefficient of friction (COF) allowed us to study different configurations of stick-slip regions and, therefore, to simulate different fretting scenarios.

^{®}was used to describe the contact behavior and the Lagrange multiplier formulation was used to define the tangential behavior of the contact pair. The surface-to-surface and finite sliding options were used to define the contact interaction.

_{y}= 0). The sides of the pads were restricted from horizontal movement in the x direction (U

_{x}= 0) and the Multiple point constraints (MPC) tie constraint was also used at the top surface of the pad to guarantee that it would not rotate due to the applied concentrated load, F.

_{reaction}). This reaction stress is obtained as:

_{axial}are obtained from experimental data (see Table 2). For this study, they are taken from the experimental set-up FF1 in Reference [25].

^{®}. In the first loading step, the top pad was pressed against the specimen surface by a normal load F = 543 N and this compressed condition was held constant until the end of the cycle. Then, both axial and reaction maximum stresses were applied to the sides of the specimen (values are presented in Table 2). Finally, in the third loading step, both axial and reaction minimum stresses were applied.

## 4. Results and Discussion

_{1}) and at leading edge side q(x

_{2}) (see Figure 3) and the peak tangential stress in the x direction, σ

_{xx,max}(see Figure 4). The influence of the mesh size on the values of the ratios between stick and slip zones sizes (c/a) is also considered here. The slip zone size, c, is obtained by measuring the position in the contact that have non-zero values of slip and the contact width, a, is obtained by the position in the x direction of the edges of the contact region, both calculated from ABAQUS

^{®}.

_{xx,max}, they seem to converge, but to a different value than the estimate from Equation (7). This is reasonable, since this equation provides only an approximate value of σ

_{xx,max}. Note that the non-dimensional parameter (c/a) also converges on the analytical solution for all values of coefficient of friction and pad radius.

#### 4.1. Stress Singularity Check: Influence of Mesh Size on Stress Components

_{rel,an}) was calculated as:

_{xx,max}, seems to converge to a different value than the estimate from Equation (7). Therefore, in order to study the convergence of the results of the, instead of considering the analytical solution as a reference, the maximum stresses between two subsequent mesh refinements were used to calculate the relative error e

_{rel}as:

#### 4.2. Fretting Fatigue Convergence Map

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Typical normalized shear traction distribution at contact interface (Q = 155.165 N, σ

_{axial}= 100 MPa, p

_{max}= 185.03 MPa, µ = 0.4 and a = 0.467 mm).

**Figure 4.**Typical normalized principal stress σ

_{xx}distribution at contact interface, obtained from finite element analysis (Q = 155.165 N, σ

_{axial}= 100 MPa, p

_{max}= 185.03 MPa, µ = 0.85 and a = 0.467 mm).

**Figure 5.**Details of the models: (

**a**) dimensions of FE fretting fatigue model with 10 mm pad radius; (

**b**) dimensions of FE fretting fatigue model with 50 mm pad radius; (

**c**) loading and (

**d**) boundary conditions.

**Figure 6.**Details of the used mesh: (

**a**) partition of model and the edges seeding (element size h varied from values: 20 µm, 10 µm, 5 µm, 2.5 µm, 1.25 µm, 0.625 µm, 0.3125 µm) and (

**b**) an illustration of the model with pad radius of 10 mm and mesh size of 2.5 µm.

**Figure 8.**Mesh convergence curves for the peak values of shear stress near the trailing edge for different pad geometries: (

**a**) Pad radius of 50 mm and (

**b**) pad radius of 10 mm and also for the peak values of shear stress near the leading edge, considering different pad geometries; (

**c**) pad radius of 50 mm and (

**d**) pad radius of 10 mm.

**Figure 9.**Contact shear traction at contact interface for different mesh sizes, pad radius and coefficients of friction: (

**a**) Pad radius of 50 mm and COF 0.3; (

**b**) pad radius of 10 mm and COF 0.3 (

**c**) pad radius of 50 mm and COF 2.0; and (

**d**) pad radius of 10mm and COF 2.0.

**Figure 10.**Mesh convergence curves for the maximum tangential stress for different cases of coefficient of friction and different pad radius: (

**a**) 50 mm and (

**b**) 10 mm.

**Figure 11.**Tangential stress at contact interface as function of normalized contact width. Results from FEA model with mesh size equal to 0.3125 µm for different pad radius: (

**a**) 50 mm and (

**b**) 10 mm.

**Figure 12.**Fretting fatigue convergence map: Stick–slip ratio (c/a) as function of the element size in the contact zone for different numerical accuracies (1%, 2% and 5%).

E | Modulus of Elasticity [GPa] | 72.1 |

ν | Poisson’s ratio | 0.33 |

σ_{0.2} | Yield Strength [MPa] | 506 ± 9 |

**Table 2.**Values of maximum and minimum σ

_{reaction}and σ

_{axial}, based on data from experimental test FF1 from Reference [26].

Steps | σ_{axial} [MPa] | σ_{reaction} [MPa] | Q [N] |
---|---|---|---|

Step 2 (maximum values) | 100 | 92.2 | 155.165 |

Step 3 (minimum values) | 10 | 17.8 | −155.165 |

**Table 3.**FEA results and analytical solution for different coefficients of friction, different pad radius and different mesh sizes at the contact surface.

Mesh size [µm] | Contact Shear Traction at Leading Side q(x_{1}) [MPa] | Contact Shear Traction at Trailing Side q(x_{2}) [MPa] | Maximum Tangential Stress σ_{xx,max} [MPa] | c/a | |||||
---|---|---|---|---|---|---|---|---|---|

R = 50 mm | R = 10 mm | R = 50 mm | R = 10 mm | R = 50 mm | R = 10 mm | R = 50 mm | R = 10 mm | ||

COF: 0.3 | 20 | 38.81 | 107.02 | 53.72 | 119.93 | 167.49 | 208.72 | 0.167 | 0.136 |

10 | 40.72 | 108.69 | 54.11 | 122.34 | 182.87 | 242.57 | 0.200 | 0.136 | |

5 | 41.01 | 110.40 | 54.23 | 123.19 | 192.96 | 268.70 | 0.206 | 0.174 | |

2.5 | 41.30 | 111.21 | 54.38 | 124.29 | 200.09 | 290.50 | 0.211 | 0.207 | |

1.25 | 41.47 | 111.44 | 54.24 | 124.33 | 205.33 | 307.81 | 0.212 | 0.211 | |

0.625 | 41.54 | 111.97 | 54.27 | 124.39 | 208.99 | 320.33 | 0.212 | 0.212 | |

0.3125 | 41.57 | 111.99 | 54.27 | 124.44 | 212.00 | 329.94 | 0.211 | 0.213 | |

Analytical | 41.29 | 112.68 | 53.99 | 124.11 | 208.35 | 342.28 | 0.218 | 0.218 | |

COF: 0.85 | 20 | 24.89 | 113.00 | 112.64 | 209.21 | 222.91 | 279.65 | 0.702 | 0.727 |

10 | 30.33 | 142.02 | 115.31 | 216.27 | 254.12 | 349.91 | 0.779 | 0.750 | |

5 | 33.31 | 146.77 | 117.18 | 226.39 | 274.64 | 398.69 | 0.788 | 0.779 | |

2.5 | 36.73 | 155.01 | 118.06 | 230.10 | 287.93 | 440.60 | 0.805 | 0.799 | |

1.25 | 37.54 | 158.98 | 118.50 | 233.62 | 297.38 | 474.40 | 0.806 | 0.804 | |

0.625 | 38.34 | 160.91 | 118.89 | 234.83 | 303.26 | 496.58 | 0.808 | 0.808 | |

0.3125 | 38.80 | 162.08 | 119.01 | 235.81 | 308.06 | 513.08 | 0.808 | 0.809 | |

Analytical | 38.09 | 163.18 | 119.20 | 235.12 | 283.90 | 507.82 | 0.811 | 0.815 | |

COF: 2.0 | 20 | 17.53 | 123.20 | 165.85 | 240.56 | 275.64 | 322.49 | 0.893 | 0.667 |

10 | 22.50 | 162.01 | 175.64 | 318.97 | 336.52 | 441.20 | 0.905 | 0.864 | |

5 | 28.59 | 195.58 | 181.98 | 327.15 | 373.08 | 537.49 | 0.910 | 0.895 | |

2.5 | 41.18 | 207.99 | 185.62 | 347.98 | 399.72 | 612.33 | 0.920 | 0.911 | |

1.25 | 44.39 | 227.64 | 187.36 | 357.64 | 417.17 | 683.24 | 0.918 | 0.917 | |

0.625 | 45.85 | 236.01 | 189.13 | 362.74 | 428.46 | 719.75 | 0.921 | 0.920 | |

0.3125 | 47.72 | 239.73 | 189.43 | 366.06 | 436.21 | 750.50 | 0.921 | 0.921 | |

Analytical | 46.52 | 242.70 | 190.72 | 368.12 | 382.10 | 725.56 | 0.925 | 0.926 |

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**MDPI and ACS Style**

Pereira, K.; Bordas, S.; Tomar, S.; Trobec, R.; Depolli, M.; Kosec, G.; Abdel Wahab, M.
On the Convergence of Stresses in Fretting Fatigue. *Materials* **2016**, *9*, 639.
https://doi.org/10.3390/ma9080639

**AMA Style**

Pereira K, Bordas S, Tomar S, Trobec R, Depolli M, Kosec G, Abdel Wahab M.
On the Convergence of Stresses in Fretting Fatigue. *Materials*. 2016; 9(8):639.
https://doi.org/10.3390/ma9080639

**Chicago/Turabian Style**

Pereira, Kyvia, Stephane Bordas, Satyendra Tomar, Roman Trobec, Matjaz Depolli, Gregor Kosec, and Magd Abdel Wahab.
2016. "On the Convergence of Stresses in Fretting Fatigue" *Materials* 9, no. 8: 639.
https://doi.org/10.3390/ma9080639