# Retardation of Fatigue Crack Growth in Rotating Bending Specimens with Semi-Elliptical Cracks

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## Abstract

**:**

_{OL}= 2.0 and R

_{OL}= 2.5 are conducted. The experimental results are compared to a crack growth assessment based on a modified NASGRO equation as well as the retardation model by Willenborg, Gallagher, and Hughes. The evaluated delay cycle number due to the overload by the experiments and the model shows a sound agreement validating the applicability of the presented approach.

## 1. Introduction

- Crack growth tests with 1:3 round bar specimens, incorporating constant amplitude loading, as well as the effect of overloads.
- Application of a yield zone model based on small-scale SEB specimen test data to assess the influence of overloads on the fatigue crack retardation of the 1:3 specimens.
- Comparison of results by experiments, modelling and evaluation of the transferability of test data from small-scale SEB specimens, with straight crack fronts to round bars containing semi-elliptical cracks.

## 2. Materials and Methods

_{th}is the threshold stress intensity range, K

_{C}is the fracture toughness, and C, m, p, and q are material constants. The factor F is calculated based on Equation (2) considering the crack opening function f, which is defined as the ratio of the crack opening and maximum value of the stress intensity factor, see [37].

_{th}acts as one input material parameter for the crack growth assessment. Therefore, not only the threshold of the long crack ΔK

_{th,lc}, but also the effective value ΔK

_{th,eff}for physically short cracks need to be considered. The transition from ΔK

_{th,eff}to ΔK

_{th,lc}by a certain value of crack extension Δa is based on crack closure effects [38] and denoted as the crack growth resistance curve for the threshold of the stress intensity range, usually abbreviated as R-curve. Details regarding the determination and limitations are provided in [39]. In [34], the R-curve is defined by Equation (3), where the parameters l

_{i}act as fictitious length scales for the build-up of the different crack closure effects.

_{res,OL}can be determined by Equation (5).

_{OL}is a dimensionless constant for the plasticity-induced residual stress intensity factor, K

_{max,OL}is the maximum stress intensity factor during the overload, Δa is the crack extension, z

_{OL}is the size of the overload influenced zone, γ

_{OL}is a material-dependent exponent, and K

_{max}is the maximum stress intensity factor of the basic load, see Equation (1).

_{OL}is calculated by α(K

_{max,OL}/σ

_{y})

^{2}with α depending on plane stress or strain condition, see [13,16]. In [35], the value z

_{OL}is evaluated based on Equation (6), where the parameters L

_{OL}and p

_{OL}are determined by statistical regression based on experiments with SEB specimens and ΔK

_{th,0}equals the long crack threshold value at R = 0.

_{eff}, which is calculated by Equation (7). One can see that K

_{res,OL}influences R

_{eff}and therefore affects the crack growth rate da/dN.

_{OL}= 1.0, γ

_{OL}= 0.37, p

_{OL}= 2.72, and L

_{OL}= 7.62 × 10

^{−4}mm are used, for details see [35]. In accordance with a preceding study [41] that focuses on the constant crack growth behavior of another commonly used steel material for railway axles, namely EA1N, the stress intensity factor for the semi-elliptical crack in round bars is analytically calculated according to [42], and furthermore summarized in [43]. As aforementioned, the surface crack length is optically measured during the experiments [32]; hence, in the following, all test results, as well as crack propagation parameters are related to the crack extension at the surface. Due to the cut-out of the specimens from real railway axle blanks, minor residual stresses up to 20 MPa are still measured, see [41]. As it is highlighted in [41] that these comparably minor residual stresses significantly affect the crack growth characteristics, the accordant residual stress values are considered within this study to properly assess the crack propagation. Further details are given in [41].

## 3. Results

#### 3.1. Constant Amplitude Tests

_{a}= 100 MPa. During the experiment, the surface crack length, 2s, is optically measured and the crack propagation test is stopped at a final crack length of 2s~18 mm. Utilizing the crack length 2s versus the accordant number of load-cycles, N, in Figure 1a, the crack propagation rate d(2s)/dN is computed. The corresponding surface stress intensity factor range ΔK

_{S}is calculated based on the procedure in [42], thereby enabling the representation of the d(2s)/dN vs. ΔK

_{S}diagram, as depicted in Figure 1b, for further comparison with the crack propagation model.

_{S}curve as presented. The fracture surface of CAL test #1 is illustrated in Figure 2. One can clearly see the initial starting notch at the top center of the picture as well as the further crack propagation area. At the final surface crack length of 2s~18 mm, the crack depth exhibits a~8 mm leading to a final ratio of a/s~0.9.

_{a}= 150 MPa, is demonstrated. Starting from the same crack length of 2s = 4 mm as for CAL test #1, the total lifetime until the final crack length of 2s~18 mm is only about 1 × 10

^{6}load-cycles, due to the increased bending load, see Figure 3a. Again, the crack propagation rate d(2s)/dN versus the surface stress intensity factor range ΔK

_{S}is evaluated, shown in Figure 3b.

#### 3.2. Overload Tests

#### 3.2.1. Overload Ratio R_{OL} = 2.0

_{OL}= 2.0 is illustrated. The surface crack length 2s over load-cycles N in Figure 5a shows that the overload is applied at 2s~9.4 mm, which equals a surface stress intensity factor range of ΔK

_{S}~17 MPa·m

^{1/2}under the base load bending stress of σ

_{a}= 100 MPa.

_{S}diagram in Figure 5b, whereas the crack growth rate after the overload at ΔK

_{S}~17 MPa·m

^{1/2}is significantly decreased. However, after a certain number of further load-cycles at the base load, denoted as delay cycles N

_{d}[44], the effect of the overload is passed and the crack propagation rate proceeds in accordance with the constant load tests. For Overload test #1, the delay cycle number is evaluated to N

_{d}~2.6 × 10

^{5}, which proves the beneficial retardation effect.

_{OL}= 2.0 under the same testing conditions. Again, the retardation effect is pronounced, leading to a delay cycle number of N

_{d}~1.7 × 10

^{5}. Compared with the Overload test #1 this value is reduced; however, it is still beneficial influence as the overload is recognizable.

_{OL}= 2.0 is shown in Figure 8. In general, the results reveal a sound agreement between the model and the experiments. The parameter set, considering RF = 0.05 exhibits a greater decrease of the retardation-affected crack growth rate, leading to a delay cycle number of N

_{d}~3.78 × 10

^{5}compared to RF = 0.10 with N

_{d}~2.1 × 10

^{5}as highlighted in the preceding paragraph. However, the applied model seems to cover both the constant amplitude as well as the overload-affected region well. A further discussion comparing the delay cycle number N

_{d}of the model to the experiments is given in Section 4.

#### 3.2.2. Overload Ratio R_{OL} = 2.5

_{OL}= 2.0, two further experiments with R

_{OL}= 2.5, denoted as Overload tests #3 and #4, are conducted. The test results are depicted in Figure 9 and Figure 10 respectively, which highlight a pronounced retardation effect in both cases. The delay cycle number of Overload test #3 is evaluated to N

_{d}= 4.7 × 10

^{5}, and for Overload test #4 to N

_{d}= 8.5 × 10

^{5}. On average, this equals an increase in N

_{d}from R

_{OL}= 2.0 to R

_{OL}= 2.5 by about a factor of three, proving the significant impact of the overload ratio R

_{OL}on the retardation effect.

_{OL}= 2.0 in Figure 8 before, the Overload tests #3 and #4 with R

_{OL}= 2.5 are again compared to the crack propagation model considering RF = 0.10 and RF = 0.05, see Figure 11. Here, the model again fits well to both experiments, whereas the retardation factor of RF = 0.10 leads to a reduced pronounced overload effect, with a final delay cycle number of N

_{d}~6.7 × 10

^{5}compared with the value of RF = 0.05 leading to N

_{d}~1.6 × 10

^{6}. A comparison of the delay cycles N

_{d}of the model and the experiments is shown in the next section.

## 4. Discussion

_{d}. Table 4 summarizes the values of N

_{d}for the Overload tests #1 and #2 with R

_{OL}= 2.0 with the results of the model, by considering a retardation factor RF = 0.10 and RF = 0.05. On average, a delay cycle number of N

_{d}~2.2 × 10

^{5}is evaluated for the experiments, which equals well the value of 2.1 × 10

^{5}of the model using RF = 0.10. Applying a factor of RF = 0.05, a significantly increased pronounced overload effect occurs, which leads to a non-conservative delay cycle number of 3.8 × 10

^{5}.

_{OL}= 2.5 is provided in Table 5. There, a mean value of of N

_{d}~6.6 × 10

^{5}is evaluated for the experiments, which again matches well to the value of 6.7 × 10

^{5}of the model using RF = 0.10. Considering RF = 0.05 leads to a non-conservative assessment with a delay cycle number of 1.6 × 10

^{6}as shown for R

_{OL}= 2.0.

_{d}between both experiments can be observed. Thereby, the N

_{d}-values are by trend in line with the crack growth behavior at the constant base load, whereby test #3, which exhibits a minor value of N

_{d}due to the overload, reveals a comparably increased crack propagation rate compared to test #4.

## 5. Conclusions

- Retardation effects, due to the overloads, significantly affect the crack growth rate leading to an enhancement of the lifetime. Considering the presented test results at overload ratios of R
_{OL}= 2.0 and R_{OL}= 2.5, the influence is more pronounced at higher R_{OL}-values. - The presented crack propagation model based on a modified NASGRO equation and considering the approach by Willenborg, Gallagher, and Hughes to cover retardation effects fits well with the conducted 1:3 round specimen overload tests. The additionally introduced retardation factor RF, which defines the maximum decrease of the crack propagation rate due to overloads, seems to exhibit a remarkable influence on the delay cycle number N
_{d}. In this study, the suggested value of RF = 0.10, which is evaluated based on preceding SEB tests, maintains a sound applicability. - As all model parameters are evaluated on the basis of small-scale SEB tests, the transferability of these values, by considering the effect of specimen size, geometry, as well as shape of the crack front, is validated based on the results in this study.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Results of amplitude load tests (CAL) test #1 at load of σ

_{a}= 100 MPa (

**a**) surface crack length vs. load-cycles; (

**b**) crack propagation rate versus stress intensity factor.

**Figure 3.**Results of CAL test #2 at load of σ

_{a}= 150 MPa (

**a**) surface crack length vs. load-cycles; (

**b**) crack propagation rate vs. stress intensity factor.

**Figure 5.**Results of Overload test #1 with R

_{OL}= 2.0 (

**a**) surface crack length versus load-cycles; (

**b**) crack propagation rate vs. stress intensity factor.

**Figure 7.**Results of Overload test #2 with R

_{OL}= 2.0 (

**a**) surface crack length versus load-cycles; (

**b**) crack propagation rate vs. stress intensity factor.

**Figure 9.**Results of Overload test #3 with R

_{OL}= 2.5 (

**a**) surface crack length versus load-cycles; (

**b**) crack propagation rate vs. stress intensity factor.

**Figure 10.**Results of Overload test #4 with R

_{OL}= 2.5 (

**a**) surface crack length versus load-cycles; (

**b**) crack propagation rate vs. stress intensity factor.

**Table 1.**Nominal chemical composition of investigated steel material in weight per cent [33].

Steel | C | Si | Mn | Cr | Mo | P | S | Fe |
---|---|---|---|---|---|---|---|---|

EA4T | 0.26 | 0.29 | 0.70 | 1.00 | 0.20 | 0.0200 | 0.007 | Balance |

**Table 2.**Nominal mechanical properties of investigated steel material [33].

Steel | f_{y} (MPa) | f_{u} (MPa) | A (%) |
---|---|---|---|

EA4T | 631 | 789 | 18.5 |

C (mm/(MPa√m)) | m (-) | p (-) | ν_{1} (-) | ν_{2} (-) | l_{1} (mm) | l_{2} (mm) | l_{F} (mm) | ΔK_{th,eff} (MPa√m) | ΔK_{th,0} (MPa√m) |

1.92 × 10^{−8} | 2.64 | 0.32 | 0.43 | 0.57 | 0.41e-3 | 1.75 | 0.01 | 2.00 | 7.35 |

Experiment | N_{d} by Experiment (-) | N_{d} by Crack Propagation Model (-) |
---|---|---|

Overload test #1 | 2.6 × 10^{5} | 2.1 × 10^{5} (RF = 0.10) and 3.8 × 10^{5} (RF = 0.05) |

Overload test #2 | 1.7 × 10^{5} |

Experiment | N_{d} by Experiment (-) | N_{d} by Crack Propagation Model (-) |
---|---|---|

Overload test #3 | 4.7 × 10^{5} | 6.7 × 10^{5} (RF = 0.10) and 1.6 × 10^{6} (RF = 0.05) |

Overload test #4 | 8.5 × 10^{5} |

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**MDPI and ACS Style**

Leitner, M.; Simunek, D.; Maierhofer, J.; Gänser, H.-P.; Pippan, R.
Retardation of Fatigue Crack Growth in Rotating Bending Specimens with Semi-Elliptical Cracks. *Metals* **2019**, *9*, 156.
https://doi.org/10.3390/met9020156

**AMA Style**

Leitner M, Simunek D, Maierhofer J, Gänser H-P, Pippan R.
Retardation of Fatigue Crack Growth in Rotating Bending Specimens with Semi-Elliptical Cracks. *Metals*. 2019; 9(2):156.
https://doi.org/10.3390/met9020156

**Chicago/Turabian Style**

Leitner, Martin, David Simunek, Jürgen Maierhofer, Hans-Peter Gänser, and Reinhard Pippan.
2019. "Retardation of Fatigue Crack Growth in Rotating Bending Specimens with Semi-Elliptical Cracks" *Metals* 9, no. 2: 156.
https://doi.org/10.3390/met9020156