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Prediction of Fatigue Cracks Using Gamma Function

Abdelfetah Moussouni
Abdelfetah Moussouni
, 
Mustapha Benachour
Mustapha Benachour
 and 
Nadjia Benachour
Nadjia Benachour
   | Nov 28, 2023
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Article Category: research article
Published Online: Nov 28, 2023
Page range: 29 - 46
DOI: https://doi.org/10.2478/fas-2022-0004
Keywords
© 2022 Abdelfetah Moussouni et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1.

Four points fatigue bending tests. (a) Schematic assembly and dimensions; (b) real assembly with electrical follow; and (c) fractured specimen.
Four points fatigue bending tests. (a) Schematic assembly and dimensions; (b) real assembly with electrical follow; and (c) fractured specimen.

Figure 2.

Servo-hydraulic machine for fatigue tests: Instron 8500.
Servo-hydraulic machine for fatigue tests: Instron 8500.

Figure 3.

Experimental fatigue life of V-notch in four-point bending specimens’ tests of 2024 T351 Al-alloy.
Experimental fatigue life of V-notch in four-point bending specimens’ tests of 2024 T351 Al-alloy.

Figure 4.

Comparison of predicted and experimental fatigue life at R = 0.1.
Comparison of predicted and experimental fatigue life at R = 0.1.

Figure 5.

Comparison of predicted and experimental fatigue life at R = 0.2.
Comparison of predicted and experimental fatigue life at R = 0.2.

Figure 6.

Comparison of predicted and experimental fatigue life at R = 0.3.
Comparison of predicted and experimental fatigue life at R = 0.3.

Figure 7.

Comparison of predicted and experimental fatigue life at R = 0.5.
Comparison of predicted and experimental fatigue life at R = 0.5.

Figure 8.

Comparison of predicted and experimental crack growth rate at R = 0.1.
Comparison of predicted and experimental crack growth rate at R = 0.1.

Figure 9.

Comparison of predicted and experimental crack growth rate at R = 0.2.
Comparison of predicted and experimental crack growth rate at R = 0.2.

Figure 10.

Comparison of predicted and experimental crack growth rate at R = 0.3.
Comparison of predicted and experimental crack growth rate at R = 0.3.

Figure 11.

Comparison of predicted and experimental crack growth rate at R = 0.5.
Comparison of predicted and experimental crack growth rate at R = 0.5.

Figure 12.

Error band scatter of predicted life at R = 0.1.
Error band scatter of predicted life at R = 0.1.

Figure 13.

Error band scatter of predicted life for at R = 0.2.
Error band scatter of predicted life for at R = 0.2.

Figure 14.

Error band scatter of predicted life at R = 0.3.
Error band scatter of predicted life at R = 0.3.

Figure 15.

Error band scatter of predicted life at R = 0.5.
Error band scatter of predicted life at R = 0.5.

Experimental loading conditions and equivalents’ initial and final crack lengths for different stress ratios.

R-ratio a0(mm) af(mm) Pmin(KN) Pmax(KN) P(KN) N(cycles) ΔK0(MP√m) ΔKf(MP√m)
0.1 3.34 7.875 0.115 1.149 1.034 382.000 5.395 22.696
0.2 3.31 7.14 0.237 1.184 0.947 569.700 4.85 16.88
0.3 3.365 7.365 0.348 1.16 0.812 547.000 4.22 15.90
0.5 2.735 6.28 1.25 2.50 1.25 240.000 5.535 15.82

Coefficient of Paris model.

R-ratio C m
0.1 1 · 10−8 3.645
0.2 1 · 10−8 3.7053
0.3 9 · 10−9 3.8712
0.5 5 · 10−8 2.9287

Mechanical properties of Al-Alloy 2024 T351.

σ0.2 σR A E G ν
363 477 12.5 74 27.82 0.33

Dimension of fatigue specimens.

Lt L l h B a0
64 50 14.5 10 10 2

Chemical composition of Al-alloy 2024 T351.

Elements Si Fe Cu Mn Mg Cr Ti Zn Pb Ni Al
% 0.105 0.159 3.97 0.449 1.5 0.05 0.018 0.109 0.056 0.02 Rest

Performance of Gamma model

R-ratio 0.1 0.2 0.3 0.5 Mean value
%Dev 4.520 3.892 1.220 5.200 3.708
Prediction ratio 0.961 0.968 0.988 0.951 0.967

The values of the constants of the adjustment curves.

R-ratio A B C D E F
0.1 −44,955 · 104 282,743 · 103 −71,01 · 103 8932.6 −566.95 15.047
0.2 −16,180 · 104 139,644 · 103 −42.59 · 103 6101.5 −424.49 12.113
0.3 −10,975 · 104 603,626 · 103 −132.9 · 103 14.714 −823.39 19.20
0.5 −56,762 · 104 384,000 · 103 −104.6 · 103 14.404 −1008.9 29.449
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