# Stress Ratio and Notch Effects on the Very High Cycle Fatigue Properties of a Near-Alpha Titanium Alloy

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## Abstract

**:**

^{10}cycles were performed on a turbine engine titanium alloy (Ti-8Al-1Mo-1V) at the stress ratio (R) of −1 with smooth specimens and at R = −1, 0.1 and 0.5 with notched specimens. As a result, with increase of fatigue life, the source of reduced fatigue life caused by multi-point surface crack initiation changes from crack propagation stage to crack initiation stage in the high cycle fatigue regime. Notch effect further promotes the degeneration of high cycle and very high cycle fatigue strength at R > −1. The bilinear model, extended from the Goodman method, can better estimate the mean stress sensitivity of this titanium alloy. The fatigue mean stress sensitivity and fatigue-creep mean stress sensitivity of this material increased with the increase of fatigue life. The new model, based on the Murakami model, can provide more appropriate predictions for notch fatigue strength.

## 1. Introduction

^{7}cycles) regime. The conventional fatigue limit is determined at 10

^{7}cycles, which can be achieved in less than 2.8 h, far below the in-service time for blades. Thus, it is necessary to study the VHCF properties of this titanium alloy in the designing of the blade.

^{7}cycles. Indeed, the coupling effects of stress ratio and notch on the HCF and VHCF properties of titanium alloys have been rarely reported.

## 2. Material and Experimental Procedure

_{p}) and the remaining 15% lamellar structure (α

_{s}+ β), where secondary α laths (α

_{s}) are randomly embedded in the β matrix. Therefore, the microstructure is of a duplex pattern.

_{t}) of the notched specimen was calculated as 1.82 by finite element method.

## 3. Experimental Results

^{5}cycles to 10

^{10}cycles as shown in Figure 3. A three-parameter equation was employed to calculate the stress plateau values of S-N curves. Its formula is expressed as:

_{a}) in this study. S

_{f}is fatigue limit, that is, the stress plateau value. N

_{f}is fatigue life. m and c are two constants. As a result, the stress plateau values of 297.4 MPa for the notched specimens at R = −1 and 161.5 MPa for the notched specimens at R = 0.1 were numerically obtained by nonlinear fitting with Equation (1). In continuous declining regions of S-N data, their S-N curves were statistically determined for a 50% survival probability. The well-known Basquin equation was used at this moment. According to the above methods, all S-N curves in the whole life region were obtained, as presented in Figure 3. Note that the unbroken specimens were treated as failure in this study, so the results are conservative.

^{9}cycles. All the fracture surfaces of notched specimens presented surface crack initiation.

_{p}grains, then coalesce to micro-crack clusters with extremely rough paths, which finally results in interior crack initiation failure in the VHCF regime [15]. During the coalescence processes in the interior of α

_{p}grains and between neighboring grains, numerous cyclic pressing results in grain refinement of the interior crack surface, which accounts for the formation of fine granules [11,15].

_{p}grains, as shown in Figure 1. The critical microstructure resulting in crack initiation is the preferred micro-texture of α

_{p}grains for this alloy [26]. The formation of facets and their coalescence process constitute the whole crack initiation stage [8,19]. Thus, by utilizing the facets presented in Figure 5d, the crack initiation region can be recognized completely by the dashed line, as depicted in Figure 5c. The shape of the dashed line totally depends on the distribution of facets; they were of irregular shape in most cases. The projected area normal to the applied stress direction of the single-point surface crack initiation region (area) was measured by this method from their SEM photographs. It should be noted that the facet feature in crack initiation region is blurry at R = −1, as shown in Figure 6. The compression and friction among upper and lower crack surfaces possibly lead to the occurrence of this morphology when pressure is applied. Hence, to ensure accuracy of measurement, only the values of area at positive stress ratios were measured. Finally, the equivalent sizes ( $\sqrt{area}$), introduced by Murakami [31], were determined by calculating the square root of the values of area and plotted in Figure 7. Obviously, the equivalent sizes increased with the increase of fatigue life and stress ratio. Similar results were reported by others [6]. For surface crack initiation, the stress intensity factor range (ΔK) at crack tip can be calculated by [31]

^{7}cycles). Similar to the morphology of the single-point surface crack initiation region, facets were detected in the crack initiation region of this failure mode, as shown in Figure 8b,c.

## 4. Discussion

#### 4.1. Multi-Point Crack Initiation Effect on Fatigue Properties

^{5}to 10

^{7}cycles, is 674.7–554.2 MPa for the notched specimen, which is significantly greater than the true stress range of 588.0–537.5 MPa for the smooth specimen, especially in relative low fatigue life. Both higher loading stress at the same stress ratio [21] and lower applied stress ratio [26] will cause fatigue crack initiation with multi-points. Thus, the multi-point crack initiation behavior prevailed in notched specimen at R = −1, while this feature was not detected in the smooth specimen at R = −1 and the notched specimen at R = 0.5, as shown in Figure 3. With increase in the fatigue life, the transition of fatigue failure mode from multi-point crack initiation to single-point crack initiation will occur [14]. This feature is consistent with the results presented in Figure 3b. Almost all specimens with multi-point crack initiation failed in the low fatigue life regime (HCF). Therefore, only HCF properties are affected by multi-point crack initiation behavior.

^{6}cycles, the crack propagation life occupies a large proportion of total life [10,32]. However, when fatigue life varies from 10

^{6}to 10

^{7}cycles, most of the total life is consumed in the crack initiation process, especially in 10

^{7}cycles (over 95%) [10,32,33]. Based on the above discussion, it can be concluded that when the total life is lower than 10

^{6}cycles, the fatigue life reduced by multi-point surface crack initiation mainly results from the crack propagation stage. However, the reduced fatigue life mainly occurs in the crack initiation process, when the total life varies between 10

^{6}cycles and 10

^{7}cycles.

#### 4.2. Mean Stress Effect on Fatigue Properties

_{m}) and stress amplitude (σ

_{a}) at a given fatigue life. On the basis of the fatigue strength data listed in Table 1, the Haigh diagram for the notched specimen in HCF and VHCF regimes is plotted in Figure 9. Goodman approximations are also shown by the dashed lines for comparison, which present a linear relationship between the fatigue strength expressed by stress amplitude at R = −1 (σ

_{a}

_{,R=−1}) and ultimate tensile strength (σ

_{UTS}):

^{7}cycles (HCF), the Goodman line still provides estimation with a good margin of safety for the smooth specimen [5,10]. However, the Goodman approximation could no longer be applied for notched specimens in the HCF regime, as shown in Figure 9. This indicates that notch effect further promotes the reduction of fatigue strength at R > −1 in the HCF and VHCF regimes.

_{a}= σ

_{m}) was chosen as the inflection point of the bilinear model, as illustrated in Figure 10. When stress ratio varies from −1 to 0, the fatigue mean stress sensitivity factor (FMSSF) is defined as:

_{a}

_{,R = 0}and σ

_{m}

_{,R = 0}are stress amplitude and mean stress at R = 0, respectively. The definition of FMSSF is completely consistent with the mean stress sensitivity factor (M) proposed by Schütz [38]. When stress ratio changes from 0 to 1, the fatigue-creep mean stress sensitivity factor (FCMSSF) is defined as:

^{10}cycles were significantly greater than the values in 10

^{7}cycles, which correspond to the traditional fatigue limit. That is to say, for fatigue design with ultra-long life, a significantly higher safety margin must be adopted than traditional fatigue design.

#### 4.3. Estimation of Fatigue Strength

_{w}) of materials containing small defects or inclusions at R = −1. For surface crack initiation, the formula is expressed by:

^{7}cycles as shown in Figure 3; no fatigue limit exists. Therefore, fatigue strength, rather than fatigue limit, should be discussed.

_{t}= 1, the new model should reduce to Equation (7). Therefore, the following possible equation may be assumed:

^{6}to 10

^{9}cycles. Meanwhile, 87.5% estimated results agreed to within 10%. In conclusion, the new model based on the Murakami model can provide more appropriate predictions for notch fatigue strength by taking stress concentration into account in the HCF and VHCF regimes.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The microstructure of the Ti-8Al-1Mo-1V alloy. The testing surface corresponds to a transversal section of fatigue specimens.

**Figure 2.**Shapes and dimensions (units: mm) of (

**a**) smooth specimen and (

**b**) notched specimen for ultrasonic fatigue tests; notch radius (R) is 1 mm.

**Figure 3.**(

**a**) S-N data and curves for the smooth specimen and the notched specimen at R = −1, and (

**b**) S-N data and curves for the notched specimen at R = −1, 0.1 and 0.5. The nominal stress amplitude was obtained by dividing the applied force by the minimum cross-sectional area. (Single-point: crack initiation with single-point; multi-points: crack initiation with multi-points; runout: no broken; the numeral at the right side of the arrow indicates the quantity of same test results.).

**Figure 4.**Typical fractography of single-point interior crack initiation without facet feature (smooth specimen, R = −1, σ

_{a}= 510 MPa, N

_{f}= 9.0692 × 10

^{8}cycles); (

**a**) the whole fracture surface; (

**b**) enlargement of the “fish-eye”; (

**c**) enlargement of the crack initiation region in the center of the “fish-eye”.

**Figure 5.**Typical fractography of single-point surface crack initiation with clear facet feature (notched specimen, R = 0.5, σ

_{a}= 140 MPa, N

_{f}= 6.048 × 10

^{7}cycles); (

**a**) the whole fracture surface; (

**b**) macro profile with crack initiation region and crack propagation region; (

**c**) enlargement of crack initiation region recognized by the facet feature and separated by the dashed line; (

**d**) enlargement of the facets, as pointed out by the arrows.

**Figure 6.**Typical fractography of single-point surface crack initiation region with blurry facet feature at R = −1, (

**a**) smooth specimen (σ

_{a}= 530 MPa, N

_{f}= 2.5939 × 10

^{6}cycles) and (

**b**) notched specimen (σ

_{a}= 320 MPa, N

_{f}= 3.4223 × 10

^{6}cycles).

**Figure 7.**The equivalent size of the single-point surface crack initiation region versus fatigue life for notched specimens.

**Figure 8.**Typical fractography of multi-point crack initiation with three origin sites of A, B and C (notched specimen, R = 0.1, σ

_{a}= 150 MPa, N

_{f}= 7.460 × 10

^{6}cycles); (

**a**) the macro profile with three crack initiation regions and crack propagation region; (

**b**) enlargement of crack initiation region B; (

**c**) enlargement of crack initiation region C.

Specimen Shape | Stress Ratio | 10^{6} | 10^{7} | 10^{8} | 10^{9} | 10^{10} |
---|---|---|---|---|---|---|

Smooth | −1 | 562.2 | 537.5 | 513.9 | 491.4 | - |

Notch | −1 | 336.0 | 304.5 | 297.4 | 297.4 | 297.4 |

0.1 | 219.2 | 174.1 | 161.5 | 161.5 | 127.7 | |

0.5 | 170.7 | 156.0 | 142.5 | 130.2 | 119.0 |

Fatigue Life | 10^{6} | 10^{7} | 10^{8} | 10^{9} | 10^{10} |
---|---|---|---|---|---|

FMSSF | 0.4357 | 0.6129 | 0.6885 | 0.6885 | 1.0868 |

FCMSSF | 4.7817 | 6.1663 | 6.6817 | 6.6817 | 8.4937 |

**Table 3.**A comparison of the estimated values of fatigue strength calculated by Equations (7) and (9) together with experimental results.

Specimen Code | N_{f} | $\sqrt{\mathit{a}\mathit{r}\mathit{e}\mathit{a}}\left(\mathsf{\mu}\mathsf{m}\right)$ | σ_{a} (MPa) | σ_{w} [Equation (7)] (MPa) | σ_{w} [Equation (9)] (MPa) | Err. [Equation (7)] (%) | Err. [Equation (9)] (%) |
---|---|---|---|---|---|---|---|

9 | 2.106 × 10^{6} | 164.0 | 150 | 189 | 138 | 26.0 | −7.8 |

1 | 3.940 × 10^{6} | 134.4 | 170 | 195 | 143 | 14.9 | −15.9 |

16 | 1.618 × 10^{7} | 181.4 | 150 | 186 | 136 | 23.9 | −9.3 |

3 | 2.820 × 10^{7} | 161.5 | 150 | 189 | 139 | 26.3 | −7.6 |

11 | 3.358 × 10^{7} | 256.0 | 135 | 175 | 128 | 30.0 | −4.9 |

6 | 3.424 × 10^{7} | 177.2 | 140 | 187 | 137 | 33.2 | −2.5 |

2 | 5.579 × 10^{7} | 163.1 | 160 | 189 | 138 | 18.2 | −13.5 |

7 | 6.048 × 10^{7} | 154.0 | 140 | 191 | 140 | 36.4 | −0.2 |

12 | 7.114 × 10^{7} | 179.3 | 135 | 186 | 136 | 37.9 | 0.9 |

5 | 1.244 × 10^{8} | 217.4 | 140 | 180 | 132 | 28.8 | −5.7 |

13 | 1.543 × 10^{8} | 236.3 | 130 | 178 | 130 | 36.8 | 0.1 |

10 | 2.423 × 10^{8} | 210.3 | 140 | 181 | 133 | 29.5 | −5.2 |

17 | 4.401 × 10^{8} | 235.5 | 130 | 178 | 130 | 36.8 | 0.2 |

4 | 6.177 ×10^{8} | 208.6 | 140 | 182 | 133 | 29.7 | −5.1 |

8 | 3.062 × 10^{9} | 196.1 | 130 | 183 | 134 | 41.1 | 3.3 |

14 | 3.289 × 10^{9} | 406.7 | 120 | 162 | 119 | 35.3 | −0.9 |

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**MDPI and ACS Style**

Yang, K.; Zhong, B.; Huang, Q.; He, C.; Huang, Z.-Y.; Wang, Q.; Liu, Y.-J.
Stress Ratio and Notch Effects on the Very High Cycle Fatigue Properties of a Near-Alpha Titanium Alloy. *Materials* **2018**, *11*, 1778.
https://doi.org/10.3390/ma11091778

**AMA Style**

Yang K, Zhong B, Huang Q, He C, Huang Z-Y, Wang Q, Liu Y-J.
Stress Ratio and Notch Effects on the Very High Cycle Fatigue Properties of a Near-Alpha Titanium Alloy. *Materials*. 2018; 11(9):1778.
https://doi.org/10.3390/ma11091778

**Chicago/Turabian Style**

Yang, Kun, Bin Zhong, Qi Huang, Chao He, Zhi-Yong Huang, Qingyuan Wang, and Yong-Jie Liu.
2018. "Stress Ratio and Notch Effects on the Very High Cycle Fatigue Properties of a Near-Alpha Titanium Alloy" *Materials* 11, no. 9: 1778.
https://doi.org/10.3390/ma11091778