# Calculation of Durability and Fatigue Life Parameters of Structural Alloys Using a Multilevel Model of Acoustic Emission Pulse Flow

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Temperature–Time Dependence of Strength

#### 1.2. Energy Activation of Destruction

#### 1.3. Structurally Sensitive Parameter $\gamma $

## 2. Materials and Methods

#### 2.1. The Evaluation of Durability Parameters

#### 2.2. Acoustic Emission

#### 2.3. Multilevel Model of Acoustic Emission Pulse Flow

- The process of breaking bonds, from the point of view of the authors, is decisive in destruction and retains its kinetics during structural rearrangements under uniform loading, which compete with acts of destruction in the field of AE signals [9];
- The activity of acoustic emission for structural steels during plastic deformation is characterized by low values, and the process itself is relatively “quiet” [65];
- From the point of view of the kinetic concept of strength, the accumulation of damage through the formation, accretion, and further growth of microcracks is a continuous process throughout most of the life and at multiple levels simultaneously—it starts to occur at low stresses due to the nature of the thermal fluctuation of the rupture of bonds at the tops of cracks [23];
- From the point of view of applying the approach to real objects, plastic deformation at diagnostic loading is unacceptable—the elastic deformation stage is taken as the determining stage for AE diagnostics.

- Kinetic—approximation of the time dependence of cumulative AE parameters by homogeneous destruction (determination of the linear section of the AE dependence in semi-logarithmic coordinates) (Table 1);
- Statistical—taking into account the stabilization of the values of amplitude, frequency, and pause distributions of AE in a temporary area of homogeneous fracture;
- A sign of elastic deformation—the accumulation of micro-damage corresponding to homogeneous destruction which occurs before the beginning of structural rearrangements during plastic deformation in the upper region of direct elastic deformation.

#### 2.4. Experimental Data

^{−6}s

^{−1}in a solution of 0.5 M H2SO4 with galvanostatic polarization applied with a cathodic current density of 5 mA/cm

^{2}, AE was monitored together with the registration of the total acoustic emission count. The moment of crack initiation corresponded to a stress of 1079 MPa. The data are shown in Figure 5a,b.

^{−4}s

^{−1}at room temperature. During the tensile test, the AE was converted by the AE sensors into an electrical signal; then, the signal was amplified by a constant rate of 40 dB and passed through a bandpass filter from 10 kHz to 2 MHz. For the analysis, in addition to cumulative counting (Figure 5e,f) of the AE, the RMS voltage, peak amplitude, and incrementation time of the AE signals were also selected.

^{−5}s

^{−1}.

^{−1}. The test temperature was room temperature.

^{−3}s

^{−1}. Acoustic emission correlated well with the volume of plastically deformed material in steel, and the size of the plastic zone was estimated during the fracture toughness test. In addition, the acoustic signal was used to determine the beginning of the formation of a stable crack.

^{−2}s

^{−1}. The AE results showed that the intensity of the signals increased with an increase in the content of the Al–5Ti–1B ligature, which was associated with the combined effect of dislocation movement and grain grinding.

## 3. Results

#### 3.1. Numerical Simulation

#### 3.2. Fatigue Life Calculation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**S–N Curve—Inconel 625 (Temperature—300 K; frequency—10 Hz; R = 0.1) [45].

**Figure A2.**S–N Curve—Steel D2 (Temperature—293 K; frequency—23 Hz; R = 0.75) [99].

**Figure A4.**S–N Curve—Ti-15V-3.0Cr-3.0Al-3.0Sn (Temperature—298 K; frequency—20 Hz). Source—Base Total material.

**Figure A5.**S–N Curve—Steel AISI 304SS (Temperature—293; frequency—50 Hz; R = 1.5) [100].

**Figure A6.**S–N Curve—Steel AISI 1080 (Temperature—293 K; frequency—30 Hz; R = −1) [101].

**Figure A7.**S–N Curve—Steel AISI 1060 (Temperature—293 K; frequency—30 Hz; R = −1) [102].

**Figure A8.**S–N Curve—A516 Grade 70 Steel (Accepted as an analogue of SA333) (Temperature—293 K; frequency—30 Hz). Source—Base Total Material.

**Figure A9.**S–N Curve—Steel AISI 304LN (Temperature—293 K; frequency—30 Hz). Source—Base Total Material.

**Figure A10.**S–N Curve—Steel AISI 304L (Temperature—700 K; frequency—40 Hz; R = −1) [103].

**Figure A11.**S–N Curve—Steel 09G2S (Temperature—293 K; frequency—0.6 Hz) [104].

**Figure A12.**S–N Curve—Steel C55 (accepted as an analogue) (Temperature—293 K; frequency—30 Hz) [105].

**Figure A13.**S–N Curve—Steel AISI 316LN (Temperature—293 K; frequency—10 Hz; R = 0.1) [106].

**Figure A14.**S–N Curve—4330V (accepted as an analogue) (Temperature—293 K; frequency—50 Hz) [107].

**Figure A15.**S–N Curve—Al 7075 alloy (Temperature—293 K; frequency—60 Hz) [107].

**Figure A16.**S–N Curve—Steel S355JR (accepted as an analogue) (Temperature—293 K; frequency—30 Hz) [105].

**Figure A17.**S–N Curve—Steel M250 (Temperature—293 K; frequency—60 Hz) [108].

**Figure A18.**S–N Curve—GJS-400-15 (Temperature—293 K; frequency—20 Hz) [109].

**Figure A19.**S–N Curve—Al 5052 alloy (Temperature—293 K; frequency—4 Hz). Source—Base Total Material.

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**Figure 2.**Graphical interpretation of the determination of the activation energy of the destruction of nickel alloy 625: (

**a**)—for creep processes; (

**b**)—for fatigue failure processes.

**Figure 5.**The total AE count and the logarithm of the total AE count over time: (

**a**) nickel alloy 625+; (

**c**) tool steel D2 (sample E3); (

**e**) AISI 304 steel with 24 h exposure; (

**g**) AISI 1060 steel; (

**i**) A572 steel of grade 50 with a deformation value of 0.072; (

**k**) M250 steel with hidden defects; (

**m**) vessels made of Ti–15V–3Al–3Cr–3Sn alloy; (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**,

**n**)—corresponding loading schedules.

**Figure 6.**Correlation of the results of the calculation of the activation energy of fracture using a multilevel model of the AE pulse flow compared with: (

**a**)—data of fatigue curves of alloys; (

**b**)—Grabar’s formula for fatigue curves; (

**c**)—results of static tests of samples; (

**d**)—Moghanlou’s formula for fatigue tests.

**Figure 7.**(

**a**) Correlation between the actual destructive stress and that calculated by the multilevel AE model; (

**b**) correlation between the structural parameter $\gamma $ calculated from the multilevel AE model and the results of static tests.

**Figure 8.**The distribution of the structural parameter γ during homogeneous fracture: (

**a**)—Weibull distribution for D2 steel samples (red—C2 sample; blue—E3 sample); (

**b**)—logarithmically normal distribution for D2 steel samples (red—E3 sample; blue—C2 sample); (

**c**)—Weibull distribution for M520 steel samples; (

**d**)—logarithmically normal distribution for M520 steel samples (blue–inherent defect, red—weld seam without defect, and gray—weld seam with defect).

**Figure 10.**S–N curves for experimental materials with marked points of calculated fatigue life: (

**a**) steel 20 with pre-cycling at 390 MPa; (

**b**) steel 20 with pre-cycling at 330 MPa; (

**c**) 15X2GMF steel with pre-cycling at 800 MPa.

Stage | Destruction Phase | Diagnostic Features of the Destruction Phase |
---|---|---|

I | Delocalized, finely dispersed inhomogeneous | d^{2}ξ/dt^{2}< 0 at σ = 0; d^{2}lnξ/dt^{2} < 0 at σ = 0; dk_{ae}/dt < 0 (dP_{U} / dt < 0); ω_{2}/ω_{1} > 1, ω_{2}/ω_{0} > 1;σ _{3} > μ; ATD* = var |

I | Delocalized, finely dispersed homogeneous | d^{2}ξ/dt^{2}= 0 at σ = const; d^{2}lnξ/dt^{2} = 0 at σ = const; dk_{ae}/dt = 0; ω_{2}/ω_{1} < 1, ω_{2}/ω_{0} < 1;σ _{3} < μ; ATD = var |

I | Localized, finely dispersed inhomogeneous | d^{2}ξ/dt^{2}< 0 at σ = 0; d^{2}lnξ/dt^{2} < 0 at σ = 0;dk _{ae}/dt < 0 (dP_{U}/dt < 0); ω_{2}/ω_{1}>1, ω_{2}/ω_{0} > 1;σ _{3} > μ; ATD = invar |

I | Delocalized, finely dispersed inhomogeneous | d^{2}ξ/dt^{2}< 0 at σ = 0; d^{2}lnξ/dt^{2} < 0 at σ = 0; dk_{ae}/dt < 0 (dP_{U} / dt < 0); ω_{2}/ω_{1} > 1, ω_{2}/ω_{0} > 1;σ _{3} > μ; ATD* = var |

II | Crack formation and propagation | d^{2}ξ/dt^{2}> 0 at σ = const; d^{2}lnξ/dt^{2} > 0 at σ = const;dk _{ae}/dt > 0 (dP_{U}/dt < 0); ω_{1}/ω_{0} > 1, ω_{2}/ω_{0} > 1; σ_{3} > μ; ATD ≈ invar |

II | Ductile rupture | d^{2}ξ/dt^{2}< 0 at σ = const; d^{2}lnξ/dt^{2} < 0 at σ = const; dk_{ae}/dt < 0 (dP_{∆t}/dt < 0); ω_{1}/ω_{0} < 1, ω_{2}/ω_{0} < 1; σ_{3} < μ; ATD ≈ invar |

**Table 2.**Results of calculation of concentration-kinetic AE strength indicators, kinetic parameters, and activation energy of destruction.

№ | Material | ${\mathit{X}}_{\mathit{A}\mathit{E}},$ ${\mathbf{s}}^{-1}$ | ${\mathit{Y}}_{\mathit{A}\mathit{E}},$ $\mathbf{M}\mathbf{P}{\mathbf{a}}^{-1}$ | $\mathit{\gamma}$ | ${\mathit{P}}_{\mathit{f}},$ $\mathbf{M}\mathbf{P}\mathbf{a}$ | ${\mathit{\vartheta}}_{\mathit{s}},$ $\frac{\mathbf{M}\mathbf{P}\mathbf{a}}{\mathbf{s}}$ | $\mathit{T},$ $\mathbf{K}$ | ${\mathit{U}}_{0\mathit{A}\mathit{E}},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ | ${\mathit{U}}_{0\mathit{S}\mathit{N}}^{1},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ | ${\mathit{U}}_{0\mathit{S}\mathit{N}}^{2},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ | ${\mathit{U}}_{0\mathit{s}\mathit{t}},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ | ${\mathit{U}}_{0\mathit{S}\mathit{N}}^{3},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |

1 | Ni alloy 625+ [69] | 0.000478 | 0.010304 | 4.45 × 10−23 | 1106 | 0.046 | 313 | 141,974 | 131,069 | 136,695 | 123,326 | |

2 | Tool steel D2 [60] | 0.007583 | 0.003936 | 1.59 × 10−23 | 685 | 1.926 | 293 | 104,990 | 118,288 | 126,203 | 101,779 | 114,182 |

3 | Tool steel D2 [60] | 0.023035 | 0.012704 | 5.14 × 10−23 | 551 | 1.813 | 293 | 114,688 | 118,288 | 99,110 | 114,092 | |

4 | Ti–15V–3Al–3Cr–3Sn [71] | 0.007166 | 0.018929 | 7.65 × 10−23 | 595 | 0.379 | 293 | 125,986 | 114,491 | 113,260 | 112,335 | 108,515 |

5 | Ti–15V–3Al–3Cr–3Sn [71] | 0.006134 | 0.013623 | 5.51 × 10−23 | 580 | 0.450 | 293 | 118,181 | 114,491 | 113,260 | 104,529 | 108,515 |

6 | Ti–6Al–4V [71] | 0.011777 | 0.012317 | 4.98 × 10−23 | 455 | 0.956 | 293 | 110,999 | 108,979 | 108,911 | 97,339 | 103,811 |

7 | AISI 304 [72] | 0.038531 | 0.010372 | 4.19 × 10−23 | 685 | 3.715 | 293 | 111,767 | 108,352 | 111,981 | 101,140 | |

8 | 0.032822 | 0.008835 | 3.57 × 10−23 | 685 | 3.715 | 293 | 109,594 | 108,352 | 101,335 | |||

9 | 0.028785 | 0.007749 | 3.13 × 10−23 | 685 | 3.715 | 293 | 108,101 | 108,352 | 101,495 | |||

10 | 0.026145 | 0.007038 | 2.85 × 10−23 | 685 | 3.715 | 293 | 107,149 | 108,352 | 101,612 | |||

11 | AISI 1060 [54] | 0.012092 | 0.006982 | 2.82 × 10−23 | 629 | 1.732 | 293 | 107,981 | 110,316 | 108,526 | 94,300 | 101,722 |

12 | AISI 1080 [54] | 0.027879 | 0.018437 | 7.45 × 10−23 | 254 | 1.512 | 293 | 106,657 | 108,352 | 111,158 | 92,984 | 102,152 |

13 | AISI 304LN [54] | 0.028509 | 0.009174 | 3.71 × 10−23 | 659 | 3,108 | 293 | 109,920 | 111,230 | 112,893 | 96,263 | 104,639 |

14 | SA333 [54] | 0.009124 | 0.0237 | 9.58 × 10−23 | 232 | 0.385 | 293 | 111,362 | 115,363 | 115,675 | 97,701 | 107,802 |

15 | AISI 304L [73] | 0.004308 | 0.003218 | 3.21 × 10−23 | 293 | 1.339 | 723 | 251,932 | 264,646 | 308,930 | 215,281 | 228,173 |

16 | 09Γ2C [74] | 0.110384 | 0.007903 | 3.20 × 10−23 | 464 | 13.967 | 293 | 100,833 | 105,078 | 112,620 | 98,555 | |

17 | 0.011341 | 0.008497 | 3.44 × 10−23 | 365 | 1.335 | 293 | 104,996 | 105,078 | 98,467 | |||

18 | K3 [74] | 0.117816 | 0.008235 | 3.33 × 10−23 | 712 | 14.306 | 293 | 106,022 | 105,596 | 106,966 | 98,827 | |

19 | 0.024145 | 0.015517 | 6.27 × 10−23 | 433 | 1.556 | 293 | 111,965 | 105,596 | 98,056 | |||

20 | AISI 316LN [51] | 0.003047 | 0.000329 | 1.33 × 10−24 | 889 | 9.264 | 293 | 101,357 | 117,953 | 120,460 | 115,817 | |

21 | 0.051016 | 0.006849 | 2.77 × 10−23 | 736 | 7.448 | 293 | 106,058 | 117,953 | 111,277 | |||

22 | 5XH3MA [75] | 0.079202 | 0.009542 | 3.86 × 10−23 | 1212 | 8.301 | 293 | 120,871 | 120,481 | 119,916 | 107,220 | 113,186 |

23 | 0.040355 | 0.012987 | 5.25 × 10−23 | 1165 | 3.107 | 293 | 131,194 | 120,481 | 117,543 | 114506 | ||

24 | 0.030321 | 0.009701 | 3.92 × 10−23 | 1167 | 3.125 | 293 | 122,617 | 120,481 | 108,966 | 113340 | ||

25 | Al Alloy 7075 [76] | 0.633642 | 0.011266 | 4.56 × 10−23 | 577 | 56.245 | 293 | 103,476 | 102,397 | 106,647 | 96021 | |

26 | High-strength low-alloy steel grade A572 мapки 50 (HSLA) [54] | 0.019052 | 0.002965 | 1.20 × 10−23 | 486 | 6.425 | 293 | 996,91 | 106,460 | 109,038 | 101256 | |

27 | 0.029453 | 0.004584 | 1.85 × 10−23 | 486 | 6,425 | 293 | 100,546 | 100725 | ||||

28 | 0.03203 | 0.004985 | 2.02 × 10−23 | 486 | 6.425 | 293 | 100,816 | 100,623 | ||||

29 | 0.036099 | 0.005619 | 2.27 × 10−23 | 486 | 6.425 | 293 | 101,274 | 100,478 | ||||

30 | 0.005005 | 0.000779 | 3.15 × 10−24 | 486 | 6.425 | 293 | 100,358 | 100,731 | ||||

31 | 0.03196 | 0.004975 | 2.01 × 10−23 | 486 | 6.425 | 293 | 100,809 | 100,626 | ||||

32 | 0.045614 | 0.0071 | 2.87 × 10−23 | 486 | 6.425 | 293 | 102,457 | 100,193 | ||||

33 | 0.043259 | 0.0071 | 2.87 × 10−23 | 486 | 6.425 | 293 | 102,153 | 100,821 | ||||

34 | Steel M250 [77] | 0.002043 | 0.006605 | 2.67 × 10−23 | 1174 | 0.309 | 293 | 130,149 | 126,814 | 132,243 | 106,847 | 119,861 |

35 | 0.002333 | 0.006924 | 2.80 × 10−23 | 1915 | 0.337 | 293 | 133,581 | 126,814 | 132,243 | 119,930 | 119,803 | |

36 | 0.003785 | 0.010223 | 4.13 × 10−23 | 1681 | 0.370 | 293 | 141,961 | 126,814 | 132,243 | 128,310 | 118,910 | |

37 | Alloy GJS-400-15 [78] | 0.005369 | 0.035255 | 1.43 × 10−22 | 315 | 0.152 | 293 | 126,306 | 112,058 | 130,205 | 104,670 | |

38 | 0.004344 | 0.028838 | 1.01 × 10−22 | 324 | 0.151 | 253 | 105,804 | 112,058 | 130,205 | 105,934 | ||

39 | Al Alloy 5052 [79] | 0.010626 | 0.006312 | 2.55 × 10−23 | 114 | 1.684 | 293 | 100,369 | 111,851 | 112,002 | 106,418 |

Sample Number | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | Weibull | Logarithmically Normal | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{k}$ | $\mathit{\lambda}$ | $\mathit{q}$ | ${\mathit{E}}_{\mathit{A}}$, % | ${\mathit{E}}_{\mathit{T}\mathit{f}},\mathit{\%}$ | ${\mathbf{\sigma}}_{\mathbf{z}}$ | $\mathsf{\mu}$ | ${\mathit{E}}_{\mathit{A}},\mathit{\%}$ | ${\mathit{E}}_{\mathit{T}\mathit{f}},\mathit{\%}$ | |||

Steel D2-1 | 38.2 | 76.5 | 3 | 36 | 0 | 17 | 25 | 0.47 | 3.05 | 13 | 24 |

Steel D2-2 | 144.1 | 219.9 | 3 | 27 | 15 | 11 | 36 | 0.29 | 3.12 | 9 | 41 |

Steel M250-1 | 3871 | 4641 | 3 | 12,5 | 0 | 6 | 39 | 0.35 | 2.3 | 5 | 38 |

Steel M250-2 | 4178 | 4345 | 3 | 18 | 0 | 4 | 51 | 0.35 | 2.7 | 4 | 55 |

Steel M250-3 | 4391 | 4688 | 3 | 12 | 7 | 3 | 50 | 0.15 | 2.7 | 4 | 30 |

Material | Spent Fatigue Life | ${\mathit{X}}_{\mathit{A}\mathit{E}},$ ${\mathbf{s}}^{-1}$ | ${\mathit{Y}}_{\mathit{A}\mathit{E}},$ $\mathbf{M}\mathbf{P}{\mathbf{a}}^{-1}$ | $\mathit{\gamma},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}\mathit{\xb7}\mathbf{M}\mathbf{P}\mathbf{a}}$ | ${\mathit{U}}_{0},$ $\frac{\mathbf{J}}{\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}}$ | ${\mathit{\sigma}}_{\mathit{m}\mathit{a}\mathit{x}},$ $\mathbf{M}\mathbf{P}\mathbf{a}$ | ${\mathit{N}}_{\mathit{f}},$ $\mathbf{c}\mathbf{y}\mathbf{c}\mathbf{l}\mathbf{e}$ |
---|---|---|---|---|---|---|---|

Steel 20 [96] | Initial | 0.01557 | 0.00924 | 22.507 | 107,160 | 390 | 3,377,507 |

0.3 | 0.01103 | 0.01295 | 31.543 | 109,893 | 2,889,932 | ||

0.5 | 0.01027 | 0.01206 | 29.373 | 109,215 | 2,988,131 | ||

0.7 | 0.00928 | 0.01090 | 26.563 | 108,357 | 3,133,187 | ||

Steel 20 [95] | Initial | 0.00674 | 0.00605 | 14.720 | 105,439 | 330 | 6,200,262 |

0.3 | 0.00250 | 0.00215 | 5.228 | 103,582 | 6,237,870 | ||

0.5 | 0.00293 | 0.00249 | 6.060 | 103,590 | 6,018,547 | ||

0.7 | 0.00123 | 0.00107 | 2.594 | 104,076 | 7,692,749 | ||

15Kh2GMF [94] | Initial | 0.00102 | 0.00152 | 3.693 | 107,336 | 800 | 23,019,230 |

0.7 | 0.00110 | 0.00169 | 4.121 | 107,615 | 23,697,629 | ||

Tensile strength sample | 0.00346 | 0.00468 | 11.397 | 112,452 | 26,313,080 |

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**MDPI and ACS Style**

Perveitalov, O.G.; Nosov, V.V.; Borovkov, A.I.; Khanukhov, K.M.; Chetvertukhin, N.V.
Calculation of Durability and Fatigue Life Parameters of Structural Alloys Using a Multilevel Model of Acoustic Emission Pulse Flow. *Metals* **2023**, *13*, 4.
https://doi.org/10.3390/met13010004

**AMA Style**

Perveitalov OG, Nosov VV, Borovkov AI, Khanukhov KM, Chetvertukhin NV.
Calculation of Durability and Fatigue Life Parameters of Structural Alloys Using a Multilevel Model of Acoustic Emission Pulse Flow. *Metals*. 2023; 13(1):4.
https://doi.org/10.3390/met13010004

**Chicago/Turabian Style**

Perveitalov, Oleg G., Viktor V. Nosov, Alexey I. Borovkov, Khanukh M. Khanukhov, and Nikita V. Chetvertukhin.
2023. "Calculation of Durability and Fatigue Life Parameters of Structural Alloys Using a Multilevel Model of Acoustic Emission Pulse Flow" *Metals* 13, no. 1: 4.
https://doi.org/10.3390/met13010004