# Crack Propagation Analysis of Compression Loaded Rolling Elements

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

- Basic model with a flat crack—model of a cylinder compressed between two steel plates, that contains one circular flat crack in the middle. Although the faces of the crack can come in contact, no friction is considered between the faces. Different combinations of dimensions were considered.
- Model with a void—the crack in this model is not flat from the start, but it starts from a three-dimensional void in the middle of the cylinder. Although it starts from the void, the crack is still considered flat and circular.
- Model with a void and friction—the same as the model with a void, but friction is considered between the faces of the crack.

#### 2.1. Basic Model with a Flat Crack

_{I}, K

_{II}and K

_{III}were determined along the whole crack front. The position on the crack front is defined by the angle γ that goes from +90° across 0° to −90° (see Figure 1 for illustration).

#### 2.2. Models with a Void and Friction

^{2}. The variable parameters were the crack length, the orientation of the crack and the dimensions of the void.

_{v}was 0.75 mm (diameter d

_{v}= 1.5 mm). The height of the void h

_{v}was set as a variable to study its effect. See Figure 4 for the illustration of the situation.

_{v}/r

_{v}. The void ratio values were considered going from 0.125 up to 1.75 (with the step of 0.25 between 0.25 and 1.75). The void is shaped such as a perfect sphere in case of h

_{v}/r

_{v}= 1. In case of h

_{v}/r

_{v}< 1, the void is an oblate spheroid; in case of h

_{v}/r

_{v}> 1, the void has a shape of a prolate spheroid. Apart from the void, the model contains a circular crack similar to the model with a flat crack. The crack lengths a considered in the model with the void were the same as in the model with a flat crack, except the 0.25 mm and 0.75 mm. It was not possible to model these lengths due to the void in the middle. The crack length started at 0.85 mm and went up to 1.75 mm.

_{II}and K

_{III}. The coefficient of friction used in the contact was 0.32. This value was chosen as a conservative estimate based on relevant data about friction coefficients of POM and PEEK materials, which are usually peeking slightly above 0.3 at room temperature [37,38,39,40]. However, the friction coefficient can significantly vary for different blends of the same polymer.

## 3. Results and Discussion

_{II}and

_{KIII}, only K

_{I}was evaluated. The values for the case of the cylinder with D × L = 6 × 6 mm

^{2}in the rolling position ρ = 0° were taken from the paper and compared to the results that were produced by the model described here. The comparison is plotted in Figure 5. The values of K

_{I}are plotted as a function of the position on the crack front described by the angle γ. The discrepancy between the results of the two models is negligible. Note that Berer et al.’s values are in the range of 0°–90°, because the one-eighth-type of symmetry was used in their model. It was possible, because the values of K

_{I}are symmetrical, as shown by the newer results. However, it would not be possible to use this type of symmetry for the evaluation of K

_{II}and K

_{III}.

#### 3.1. Model with a Flat Crack—Results and Discussion

^{2}and loading force F = 350 N, unless stated otherwise. The results are plotted in Figure 6 (K

_{I}), Figure 7 (K

_{II}) and Figure 8 (K

_{III}) for one of the simulated crack lengths—1.25 mm. 3D plots were chosen to visualize the values of stress intensity factors depending on both, the position on the crack front γ and the overall rolling orientation ρ of the crack. The results were also plotted in the form of 2D plots, where the crack length a and position on the crack front γ were fixed and the stress intensity factors are plotted as a function of the rolling orientation (angle ρ). Many 2D plots had to be created to illustrate the whole situation, because of many possible combinations of parameters (crack length and position on the crack front). The 2D plots are not included in the text of this paper for the sake of clarity. The most important 2D plots are included in Appendix A.

_{I}(Figure 6) shows that the K

_{I}values reach their maximum at the beginning and at the end of the turn of the cylinder. During the turn from 0° towards 180°, the values decrease up to the point when the two crack faces come into contact. The crack stays closed until the cylinder comes into a position where the opening stress starts acting on the crack again. The K

_{I}values are zero when the crack faces are in contact in the model. A simulation without the contact of crack faces was also carried out to investigate the exact moment of the crack closing and opening. If no contact is defined between the crack faces, the K

_{I}values become negative in the part of the cycle where the crack is closed (the negative values are also plotted in Figure 6). Negative values of K

_{I}cannot occur in reality, but this kind of simulation helps to evaluate the cycle of K

_{I}and its asymmetry, which can be important for a later use in lifetime estimations and for experimental testing of such a situation.

_{II}and K

_{III}, are higher in magnitude compared to the K

_{I}in terms of maximum values. On the beginning of the turn, both K

_{II}and K

_{III}are zero along the whole crack front, as the crack is not subjected to shear loading at all. However, the conditions change with the turning. The middle of the crack (γ = 0°) is subjected to mode III type of loading and the mode II does not appear here at all during the turn, whereas the crack front ends (γ = 90° and −90°) develop mutually opposite values of K

_{II}during the cycle, and K

_{III}remains equal to zero. In between these positions, the K

_{II}and K

_{III}values follow different sine patterns. As the crack becomes perpendicular to the direction of loading (ρ = 90°), both of the shear modes disappear again. Then, in the following part of the turn, the values of K

_{II}and K

_{III}appear again in the same places on the crack front, but with opposite signs—see Figure 7 and Figure 8.

_{I}reaches the maximum values, both K

_{II}and K

_{III}are zero. There are short intervals at the beginning and the end of the turn, when the crack is open and loaded by a combination of K

_{I}and K

_{III}(for the position of γ = 0°) or K

_{I}and K

_{II}(for γ = 90° and −90°). Between these, there are intervals in which the K

_{II}and K

_{III}are non-zero and even reach their maxima (or minima) and the crack is closed (K

_{I}is zero). This means that the crack faces are being forced against each other and into one of the shear modes at the same time. It is quite likely that heat is generated by the friction of the crack faces in these intervals, which can have an influence on the crack propagation rate [31,32,41].

_{max}to describe the whole cycle. In Figure 9, these values are plotted as a function of the normalized crack length a/W (where a is the crack length and W corresponds to the radius or half of the length of the whole cylinder depending on position γ = 90° (−90°) or γ = 0°, respectively (see Figure 1). It is important to note here, that the rolling position ρ, at which the maxima and minima of K

_{I}and K

_{III}are reached, are constant with the growing crack length a. The maximum of K

_{I}can be always found at ρ = 0° and ρ = 180°, the (theoretical) minimum at ρ = 90°. The maximum of K

_{III}stays at ρ = 135° and minimum at ρ = 45°. However, the position where the K

_{II}reaches its extreme (minimum or maximum depending on the position, if γ = 90° or −90°) gradually shifts from 45° towards lower values of ρ with the crack length increasing and similarly the position of the other extreme shifts from 135° towards higher values. The shift can be observed in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6 in the Appendix A. The cause of this is most likely that the crack becomes more influenced by a complicated stress state in the vicinity of the contact with the loading plates, which manifests itself the most at the positions γ = 90° and −90°, where K

_{II}reaches its maxima and minima. This shift in the position does not influence the characterization of the stress intensity factor cycles using the K

_{max}values though.

_{min}to the maximum value K

_{max}of the cycle.

_{II}is 0. Analogically to the mode II, the R-ratio for the mode III is equal to −1, apart from the 90°and −90° positions.

_{max}values were determined for more combinations of dimensions D × L. The considered diameters D were 3, 4, 5, 6 mm, the considered lengths L were 3, 4, 5, 6, 9 and 12 mm. The entire range of crack lengths, as it is specified in Section 2, was considered for the 6 × 6 mm

^{2}type of cylinder only. Only some crack lengths were considered for the other combinations of length and diameter. These crack lengths were chosen with respect to the dimensions of the particular combination, because some of the lengths did not fit in the particular combination.

_{max}were then fitted with parametric functions that define the dependency of the stress intensity factors on the crack length and are generalized with respect to the loading force and dimensions of the cylinder. The fits were carried out for every point on the crack face, where the stress intensity factor reaches its maximum during the turn—this means there is one fit for the K

_{IImax}in the γ = 90° position and one for the K

_{IIImax}in the γ = 0° position. Two fits for the K

_{Imax}were made, one for the 90° position and another for the 0° position, because the difference between the maxima in these two positions is not very pronounced, although technically the global maximum is reached only in the 0° position. The position γ on the crack front is indicated by respective indices.

_{Imax}

_{90°}and K

_{Imax}

_{0°}are plotted in Figure 11a,b, respectively. The equation describing the dependencies are the following:

_{I}

_{90°}and Y

_{I}

_{0°}are the dimensionless shape functions that have been found in the following form:

^{1/2}, which is the typical unit used for the stress intensity factors. This is ensured by the factor of $\sqrt{{10}^{3}}$ in the denominator of both equations.

_{Imax}

_{0°}and Y

_{Imax}

_{90°}, the difference is usually not more than 6%, but for some combinations it goes up to 30% (especially when there is a larger difference between D and L of the cylinder).

_{I}properly, the K

_{Imin}values are also needed, because the R-ratio does not stay constant for the K

_{I}cycle during the crack propagation (as illustrated in Figure 10). Even though the K

_{Imin}values are only theoretical, because in practice the crack is closed and the stress intensity factor is equal to zero, knowing these values makes it possible to describe the whole cycle in detail and most importantly to determine the precise moments of the crack closing and opening. The K

_{Imin}functions were created in the same manner as the K

_{Imax}functions above. The shape function fits are plotted in Figure 12a,b. The equations follow:

_{Imin}

_{90°}and Y

_{Imin}

_{0°}are the dimensionless shape functions that have been found in the following form:

_{IImax 90°}and K

_{IIImax 0°}values. The fits for these values are plotted in Figure 13a,b, respectively. The generalized K

_{IImax}and K

_{IIImax}are much less scattered, which means that the parametric function provides a very good estimation of the real stress intensity factor values. The parametric functions have a similar form to the previous functions of K

_{Imax}. The equations are the following:

_{II}

_{90°}and Y

_{III}

_{0°}were found to be the following:

^{1/2}.

#### 3.2. Models with Voids and Friction—Results and Discussion

^{2}and with a spheroidal void in the middle. The radius of the void r

_{v}was 0.75 mm and the height of the void h

_{v}was variable. In the following, the void dimensions are described by the void ratio h

_{v}/r

_{v}.

_{v}. Figure 14a shows the K

_{Imax}values determined for the crack front positions γ = 90° (−90°) and γ = 0°. For the void ratio (h

_{v}/r

_{v}) of 0.125, the K

_{Imax}values in both γ positions are the same as for the flat crack with the same crack length (a = 0.85 mm). After that, K

_{Imax}

_{90°}increases with increasing void ratio until the ratio of 1.75, where it is approximately three times higher. K

_{Imax}

_{0°}on the other hand, decreases to lower values than those calculated for the flat crack until it becomes zero. This is caused by the decrease in stiffness of the cylinder, when the void gets more prolate (h

_{v}/r

_{v}> 1)—the crack then opens more in the positions of 90° and −90°, and less in the 0° position. Additionally, the values of stress intensity factors in the positions of 90° and −90° on the crack front are heavily influenced by the contact zone stress field.

_{Imax}

_{90°,0°}for the case of a flat crack (without the void) of the same length a = 0.85 mm is plotted as a solid line for comparison. Plotting this value as a solid line was chosen for the plot to be clearer, but it is technically incorrect, because the value would normally show as a point at r

_{v}/h

_{v}= 0.

_{IImax}for the γ = 90° (−90°) position and the K

_{IIImax}values for the γ = 0° position are shown in Figure 14b. The values of K

_{IImax}

_{90°}and K

_{IIImax}

_{0°}for the case of a flat crack of the same length a = 0.85 mm are plotted as solid lines for comparison. The K

_{IImax}

_{90°}values for the void ratio of 0.125 are slightly lower than the K

_{IImax}

_{90°}values for the flat crack length with the same crack length—by about 11%. With increasing void ratio, the K

_{IImax}

_{90°}values decrease quite rapidly. For the void ratio of 1.75, they are approximately seven times lower than the values without the void. This is exactly the opposite tendency compared to K

_{Imax}

_{90°}in the same γ position. K

_{IIImax}

_{0°}shows only a weak dependency on the void ratio. The K

_{IIImax}

_{0°}for the void ratio of 0.125 is the same as for the flat crack. For the highest simulated void ratio of 1.75, the K

_{IIImax}

_{0°}values are only slightly higher than those for the same flat crack length (by about 10%).

_{max}from the flat crack model compared to the values from the models with the voids. The results agree with previous observations on the dependencies on the varying void ratio. The presence of the void increases the K

_{I}in the 90° position and decreases the K

_{I}in the 0° position. The K

_{II}is decreased substantially by the void, whereas K

_{III}is subject to a mild increase. One feature in the results is common for all the stress intensity factors (no matter the loading mode nor the positions on the crack front): with the crack propagating further from the void, the stress intensity factors are less influenced by the presence of the void. This is given mainly by the stress concentration effect of the void, which is most important for short cracks initiated from void. The influence on the stress intensity factors for the shorter cracks is more pronounced with higher void ratios h

_{v}/r

_{v}.

_{I}has remained unchanged compared to the model with a void without friction. It is obvious, since the friction between crack faces cannot influence the tensile opening mode. The overall nature of the cycles of K

_{II}and K

_{III}with friction is also very similar to the previous model without friction. The friction affects mostly those values close to the positions of the closed crack—around the angle ρ = 90°. The difference between the values is approximately 5–10%. This means that friction does not significantly influence the overall K

_{max}functions, which can be seen also from the plot in Figure 17 that illustrates the overall influence of the friction on K

_{IImax}and K

_{IIImax}values during the crack propagation. There is again a slight shift in the positions where K

_{II}and K

_{III}cycles reach their maxima and minima with the growing crack (observable in Figure A7, Figure A8, Figure A9 and Figure A10).

## 4. Conclusions

_{I}, K

_{II}and K

_{III}, on the crack front. Originally, it was assumed that the opening mode characterized by the K

_{I}was the dominant mode, but it was found that the crack was actually closed with the load pressing the faces together during most of the cycle. Additionally, the maximum values of the shear mode stress intensity factors, K

_{II}and K

_{III}, were higher than the maximum reached by K

_{I}.

_{Imax}, K

_{IImax}and K

_{IIImax}during the rotation. These equations contain the dimensions of the cylinders and can be used to quickly describe the crack tip stress situation in the cylinder upon crack propagation without having to carry out a lengthy FEM simulation.

_{I}in the position γ = 90° (and −90°). Contrary to that, the K

_{I 0°}and K

_{II}substantially decreased in the presence of the void. The K

_{III}remains almost unchanged even in cases with a rather large void. The common observation was that with the crack propagating further from the void, the void influence decreased up to a point where the stress intensity factor values were the same as for the flat crack.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{I}in the crack opening mode I, if the faces of cracks are pressed against each other. In reality, the stress intensity factor in such a situation is 0 and it cannot be negative. The values are plotted to illustrate the instances, in which the crack closes and later opens again.

**Figure A1.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm, load F = 350 N, a = 0.25 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

**Figure A2.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 0.75 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

**Figure A3.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 0.85 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

**Figure A4.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.00 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

**Figure A5.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.25 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

**Figure A6.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.75 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°.

## Appendix B

**Figure A7.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 0.85 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°. Model including friction.

**Figure A8.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.00 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°. Model including friction.

**Figure A9.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.25 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°. Model including friction.

**Figure A10.**Course of stress intensity factors during rotation in the cylinder D × L = 6 × 6 mm

^{2}, load F = 350 N, a = 1.75 mm, position on the crack front; (

**a**) γ = 0°; (

**b**) γ = −90°. Model including friction.

## References

- Avalle, M.; Romanello, E. Tribological characterization of modified polymeric blends. Procedia Struct. Integr.
**2018**, 8, 239–255. [Google Scholar] [CrossRef] - Harrass, M.; Friedrich, K.; Almajid, A. Tribological behavior of selected engineering polymers under rolling contact. Tribol. Int.
**2010**, 43, 635–646. [Google Scholar] [CrossRef] - Berer, M.; Mitev, I.; Pinter, G. Finite element study of mode I crack opening effects in compression-loaded cracked cylinders. Eng. Fract. Mech.
**2017**, 175, 1–14. [Google Scholar] [CrossRef] - Berer, M.; Major, Z. Characterization of the global deformation behaviour of engineering plastics rolls. Int. J. Mech. Mater. Des.
**2010**, 6, 1–9. [Google Scholar] [CrossRef] - Berer, M.; Major, Z. Characterisation of the Local Deformation Behaviour of Engineering Plastics Rolls. Strain
**2011**, 48, 225–234. [Google Scholar] [CrossRef] - Bonniot, T.; Doquet, V.; Mai, S.H. Mixed mode II and III fatigue crack growth in a rail steel. Int. J. Fatigue
**2018**, 115, 42–52. [Google Scholar] [CrossRef] - Bold, P.; Brown, M.; Allen, R. Shear mode crack growth and rolling contact fatigue. Wear
**1991**, 144, 307–317. [Google Scholar] [CrossRef] - Liu, H. Material Modelling for Structural Analysis of Polyethylene. Master’s Thesis, University of Waterloo, Waterloo, ON, Canada, 2007. [Google Scholar]
- Liu, Y.; Stratman, B.; Mahadevan, S. Fatigue crack initiation life prediction of railroad wheels. Int. J. Fatigue
**2006**, 28, 747–756. [Google Scholar] [CrossRef] - Avanzini, A.; Donzella, G.; Mazzù, A.; Petrogalli, C. Wear and rolling contact fatigue of PEEK and PEEK composites. Tribol. Int.
**2013**, 57, 22–30. [Google Scholar] [CrossRef] - Kadin, Y.; Rychahivskyy, A. Modeling of surface cracks in rolling contact. Mater. Sci. Eng. A
**2012**, 541, 143–151. [Google Scholar] [CrossRef] - Berer, M.; Major, Z.; Pinter, G. Elevated pitting wear of injection molded polyetheretherketone (PEEK) rolls. Wear
**2013**, 297, 1052–1063. [Google Scholar] [CrossRef] - Koike, H.; Kida, K.; Honda, T.; Mizobe, K.; Oyama, S.; Rozwadowska, J.; Kashima, Y.; Kanemasu, K. Observation of Crack Propagation in PEEK Polymer Bearings under Water-Lubricated Conditions. Adv. Mater. Res.
**2012**, 566, 109–114. [Google Scholar] [CrossRef] - Berer, M.; Pinter, G. Determination of crack growth kinetics in non-reinforced semi-crystalline thermoplastics using the linear elastic fracture mechanics (LEFM) approach. Polym. Test.
**2013**, 32, 870–879. [Google Scholar] [CrossRef] - Berer, M.; Pinter, G.; Feuchter, M. Fracture mechanical analysis of two commercial polyoxymethylene homopolymer resins. J. Appl. Polym. Sci.
**2014**, 131, 1–15. [Google Scholar] [CrossRef] - Furmanski, J.; Pruitt, L.A. Peak stress intensity dictates fatigue crack propagation in UHMWPE. Polymer
**2007**, 48, 3512–3519. [Google Scholar] [CrossRef] - Favier, V.; Giroud, T.; Strijko, E.; Hiver, J.; G’Sell, C.; Hellinckx, S.; Goldberg, A. Slow crack propagation in polyethylene under fatigue at controlled stress intensity. Polymer
**2002**, 43, 1375–1382. [Google Scholar] [CrossRef] - Pruitt, L.; Sreekanth, P.R.; Badgayan, N.; Sahoo, S. Fatigue of Polymers. In Reference Module in Materials Science and Materials Engineering; Elsevier BV: Amsterdam, The Netherlands, 2017; pp. 1–15. [Google Scholar]
- Harris, J.S.; Ward, I.M. Fatigue-crack propagation in vinyl urethane polymers. J. Mater. Sci.
**1973**, 8, 1655–1665. [Google Scholar] [CrossRef] - Kanters, M.J.; Kurokawa, T.; Govaert, L.E. Competition between plasticity-controlled and crack-growth controlled failure in static and cyclic fatigue of thermoplastic polymer systems. Polym. Test.
**2016**, 50, 101–110. [Google Scholar] [CrossRef] - Kanters, M.J.; Stolk, J.; Govaert, L.E. Direct comparison of the compliance method with optical tracking of fatigue crack propagation in polymers. Polym. Test.
**2015**, 46, 98–107. [Google Scholar] [CrossRef] - Arbeiter, F.; Spoerk, M.; Wiener, J.; Gosch, A.; Pinter, G. Fracture mechanical characterization and lifetime estimation of near-homogeneous components produced by fused filament fabrication. Polym. Test.
**2018**, 66, 105–113. [Google Scholar] [CrossRef] - Hutař, P.; Ševčík, M.; Náhlík, L.; Pinter, G.; Frank, A.; Mitev, I. A numerical methodology for lifetime estimation of HDPE pressure pipes. Eng. Fract. Mech.
**2011**, 78, 3049–3058. [Google Scholar] [CrossRef] - Frank, A.; Arbeiter, F.J.; Berger, I.J.; Hutař, P.; Náhlík, L.; Pinter, G. Fracture Mechanics Lifetime Prediction of Polyethylene Pipes. J. Pipeline Syst. Eng. Pr.
**2019**, 10, 04018030. [Google Scholar] [CrossRef] - Frank, A.; Hutař, P.; Pinter, G. Numerical Assessment of PE 80 and PE 100 Pipe Lifetime Based on Paris-Erdogan Equation. Macromol. Symp.
**2012**, 311, 112–121. [Google Scholar] [CrossRef] - Benhamena, A.; Bouiadjra, B.B.; Amrouche, A.; Mesmacque, G.; Benseddiq, N.; Benguediab, M. Three finite element analysis of semi-elliptical crack in high density poly-ethylene pipe subjected to internal pressure. Mater. Des.
**2010**, 31, 3038–3043. [Google Scholar] [CrossRef] - Arbeiter, F.; Trávníček, L.; Petersmann, S.; Dlhý, P.; Spoerk, M.; Pinter, G.; Hutař, P. Damage tolerance-based methodology for fatigue lifetime estimation of a structural component produced by material extrusion-based additive manufacturing. Addit. Manuf.
**2020**, 36, 101730. [Google Scholar] [CrossRef] - Puigoriol-Forcada, J.M.; Alsina, A.; Salazar-Martín, A.G.; Gomez-Gras, G.; Pérez, M.A. Flexural fatigue properties of polycarbonate fused-deposition modelling specimens. Mater. Des.
**2018**, 155, 414–421. [Google Scholar] [CrossRef] - Jones, R.; Kinloch, A.; Michopoulos, J.; Brunner, A.; Phan, N. Delamination growth in polymer-matrix fibre composites and the use of fracture mechanics data for material characterisation and life prediction. Compos. Struct.
**2017**, 180, 316–333. [Google Scholar] [CrossRef] - Evans, J.W.; Sinha, K. Applications of fracture mechanics to quantitative accelerated life testing of plastic encapsulated microelectronics. Microelectron. Reliab.
**2018**, 80, 317–327. [Google Scholar] [CrossRef] - Gosch, A.; Berer, M.; Hutař, P.; Slávik, O.; Vojtek, T.; Arbeiter, F.J.; Pinter, G. Mixed Mode I/III fatigue fracture characterization of Polyoxymethylene. Int. J. Fatigue
**2020**, 130, 105269. [Google Scholar] [CrossRef] - Gosch, A.; Arbeiter, F.J.; Berer, M.; Vojtek, T.; Hutař, P.; Pinter, G. Fatigue characterization of polyethylene under mixed mode I/III conditions. Int. J. Fatigue
**2021**, 145, 106084. [Google Scholar] [CrossRef] - Schrader, P.; Gosch, A.; Berer, M.; Marzi, S. Fracture of Thin-Walled Polyoxymethylene Bulk Specimens in Modes I and III. Materials
**2020**, 13, 5096. [Google Scholar] [CrossRef] - ANSYS Help Release 2020 R2; ANSYS Inc.: Canonsburg, PA, USA, 2020.
- Shih, C.F.; Moran, B.; Nakamura, T. Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract.
**1986**, 30, 79–102. [Google Scholar] [CrossRef] - Ingraffea, A.R.; Manu, C. Stress-intensity factor computation in three dimensions with quarter-point elements. Int. J. Numer. Methods Eng.
**1980**, 15, 1427–1445. [Google Scholar] [CrossRef] - Lind, J.; Lindholm, P.; Qin, J.; Kassman, R. Friction and wear studies of some peek materials. Tribologia
**2015**, 33, 20–28. [Google Scholar] - Chaudri, A.M.; Suvanto, M.; Pakkanen, T.T. Non-lubricated friction of polybutylene terephthalate (PBT) sliding against polyoxymethylene (POM). Wear
**2015**, 342–343, 189–197. [Google Scholar] [CrossRef] - Chen, J.; Cao, Y.; Li, H. Investigation of the friction and wear behaviors of polyoxymethylene/linear low-density polyethylene/ethylene-acrylic-acid blends. Wear
**2006**, 260, 1342–1348. [Google Scholar] [CrossRef] - Hoskins, T.; Dearn, K.; Chen, Y.; Kukureka, S. The wear of PEEK in rolling–sliding contact—Simulation of polymer gear applications. Wear
**2014**, 309, 35–42. [Google Scholar] [CrossRef][Green Version] - Chen, Y.T.; Liu, K.X. Crack propagation in viscoplastic polymers: Heat generation in near-tip zone and viscoplastic cohesive model. Appl. Phys. Lett.
**2015**, 106, 061908. [Google Scholar] [CrossRef]

**Figure 1.**The situation of the bearing cylinder with an internal defect, compressed between two steel plates and rolling.

**Figure 2.**The mesh of the whole symmetrical FEM model of the bearing cylinder. The refined area in the vicinity of the crack tip is pictured in the detail view (note the special crack tip elements in the middle).

**Figure 3.**Schematic illustration of the changing orientation of the crack during rolling and the angle ρ that describes the rolling position.

**Figure 5.**Comparison of K

_{I}values from Berer et al. [3] and from the newer model described in this paper. The values from the newer model were estimated by domain integral (solid lines) and node deformations (separate points).

**Figure 6.**3D plot of K

_{I}as a function of both, the circumferential position on the crack front γ and the rolling orientation of the crack ρ, crack length a = 1.25 mm. The upper surface reflects the situation with contact defined between crack faces (highlighted by solid red line). The lower surface are theoretical negative values of K

_{I}acquired from a model where no contact was defined between crack faces (highlighted by dashed red line).

**Figure 7.**3D plot of K

_{II}as a function of both, the circumferential position on the crack front γ and the rolling orientation of the crack ρ, crack length a = 1.25 mm.

**Figure 8.**3D plot of K

_{III}as a function of both, the circumferential position on the crack front γ and the rolling orientation of the crack ρ, crack length a = 1.25 mm.

**Figure 9.**Maximum stress intensity factors in the rolling cycle for different crack tip positions γ depending on the normalized crack length. The indices 0° and 90° indicate the position γ at the crack front.

**Figure 10.**The R-ratio for different loading modes at different positions on the crack front during crack growth (

**left**) with a schematic illustration of loading cycles with different R-ratios (

**right**). The indices 0° and 90° indicate the position γ at the crack front.

**Figure 11.**(

**a**) Parametric functions fitting the values of Y

_{Imax 90°}; (

**b**) Parametric functions fitting the values of Y

_{Imax 0°}.

**Figure 12.**(

**a**) Parametric functions fitting the values of Y

_{Imin}

_{90°}; (

**b**) Parametric functions fitting the values of Y

_{Imin}

_{0°}.

**Figure 13.**(

**a**) Parametric functions fitting the values of Y

_{IImax}

_{90°}; (

**b**) Parametric functions fitting the values of Y

_{IIImax}

_{0°}.

**Figure 14.**(

**a**) Change of K

_{Imax}in different crack tip positions depending on the varying void ratio h

_{v}/r

_{v}; (

**b**) Change of K

_{IImax}and K

_{IIImax}depending on the void ratio. Both plots contain the value of K

_{Imax}

_{90°,0°}, K

_{IImax}

_{90°}and K

_{IIImax}

_{0°}for the case of flat crack a = 0.85 mm plotted as a solid line for comparison.

**Figure 15.**K

_{max}for different crack tip positions depending on the normalized crack length—comparison of the model with Table 0. mm, h

_{v}/r

_{v}= 0.25 or 1).

**Figure 16.**Comparison of stress intensity factor values for the model with void without friction and with friction: (

**a**) D × L = 6 × 6 mm

^{2}, a = 1.25 mm; γ = 0°; (

**b**) D × L = 6 × 6 mm

^{2}, a = 1.25 mm, γ = −90°.

**Figure 17.**K

_{IImax}and K

_{IIImax}depending on the normalized crack length; comparison of model with void without friction and model with void with friction; void ratio h

_{v}/r

_{v}= 0.25, r

_{v}= 0.75 mm.

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## Share and Cite

**MDPI and ACS Style**

Dlhý, P.; Poduška, J.; Berer, M.; Gosch, A.; Slávik, O.; Náhlík, L.; Hutař, P. Crack Propagation Analysis of Compression Loaded Rolling Elements. *Materials* **2021**, *14*, 2656.
https://doi.org/10.3390/ma14102656

**AMA Style**

Dlhý P, Poduška J, Berer M, Gosch A, Slávik O, Náhlík L, Hutař P. Crack Propagation Analysis of Compression Loaded Rolling Elements. *Materials*. 2021; 14(10):2656.
https://doi.org/10.3390/ma14102656

**Chicago/Turabian Style**

Dlhý, Pavol, Jan Poduška, Michael Berer, Anja Gosch, Ondrej Slávik, Luboš Náhlík, and Pavel Hutař. 2021. "Crack Propagation Analysis of Compression Loaded Rolling Elements" *Materials* 14, no. 10: 2656.
https://doi.org/10.3390/ma14102656