On the application of numerical differentiation methods to the determination of the fatigue crack growth rate

Using a sample of test results from 68 compact eccentric tensile specimens made of titanium alloys, nickel alloys and steel, the effect of the choice of numerical differentiation method (secant method and the method of differential polynomials on three, five and seven points) used to calculate the fatigue crack growth rate on characteristics of the linear section of the kinetic diagram of the fatigue failure. The purpose of the study is to determine the advantages, disadvantages and consistent patterns of the considered methods. The coefficient of determination R 2, integral criterion χ which characterizes the difference between the predicted and actual number of cycles corresponding to the section of stable crack growth, and correlation between the logarithms of the Paris constants for alloys of the same class were used as criteria for the correct choice of the method of numerical differentiation. The main results and conclusions of the study: the use of the method of differential polynomials over three points compared to the secant method slightly increases the correlation between the logarithms of the fatigue crack growth rate and the range of the stress intensity factor (an increase in R 2) and increases the difference between the calculated and experimental number of cycles corresponding to stable crack growth (an increase in χ). However, when determining the fatigue crack growth rate by the method of differential polynomials for five and seven points, a more significant smoothing of the experimental data is observed, accompanied by a significant increase in R 2 and a decrease in χ; proximity to zero of the integral accuracy parameter χ is a necessary but not sufficient criterion for good agreement between the test result and the mathematical model that describes it, while the combination of parameters χ and R 2 uniquely forms this criterion; the choice of the method of numerical differentiation does not affect the correlation of the logarithms of the constants of the Paris equation.

1. Kablov E. N., Ospennikova O. G., Bazyleva O. A. Materials for highly heat-loaded parts of gas turbine engines / Vestn. MGTU im. N. E. Baumana. Ser. Mashinostr. 2011. N SP2. P. 13 – 19 [in Russian].

2. Bondarenko Yu. A. Trends in the development of high-temperature metallic materials and technologies in the creation of modern aircraft gas turbine engines / Aviats. Mater. Tekhnol. 2019. N 2(55). P. 3 – 11 [in Russian]. DOI: 10.18577/2071-9140-2019-0-2-3-11

3. Grin’ E. A. Testing materials for corrosion-cyclic crack resistance in water at elevated temperatures / Industr. Lab. Mater. Diagn. 2010. Vol. 76. N 1. P. 57 – 60 [in Russian].

4. Savkin A. N., Andronik A. V., Badikov K. A., Sedov A. A. Study of the Kinetics of Fatigue Crack Growth in Steels Depending on the Nature of Variable Loading / Industr. Lab. Mater. Diagn. 2018. Vol. 84. N 3. P. 43 – 51 [in Russian]. DOI: 10.26896/1028-6861-2018-84-3-43-51

5. Ye H., Huang R., Zhou Y., Liu J. Calibration of Paris law constants for crack propogation analysis of damaged steel plates strengthened with prestressed CFRP / Theoretical and Applied Fracture Mechanics. 2022. Vol. 117. Article 103208. DOI: 10.1016/j.tafmec.2021.103208

6. Wang Y., Binuad N., Gogu Ch., et al. Determination of Paris’s law constants and crack length evolution via Extended and Unscented Kalmar filter: An application to aircraft fuselage panels / Mechanical Systems and Signal Processing. 2016. Vol. 80. P. 262 – 281. DOI: 10.1016/j.ymssp.2016.04.027

7. Savkin A. N., Andronik A. V., Korradi R. Method for determining the coefficients of the crack growth rate equation under cyclic loading / Industr. Lab. Mater. Diagn. 2016. Vol. 82. N 1. P. 57 – 63 [in Russian].

8. Golubovskii E. R., Volkov M. E., Emmauskii N. M. A method for determination on the boundarics of the stage of steady fatigue crack growth and parameters of Paris equation / Indust Lab. Mater Diagn. 2019. Vol. 85. N 9. P. 66 – 74 [in Russian]. DOI: 10.26896/1028-6861-2019-85-9-66-74 9

9. . Clark W. G., Hudak S. Variability in Fatigue Crack Growth Rate Testing / Journal of Testing and Evaluation. 1975. Vol. 3. N 6. P. 454 – 476.

10. Norman E. Dowling. Mechanical Behavior of Materials. Pearson Education Limited. 11. Klysz S., Leski A. Good Practice for Fatigue Crack Growth Curves Description / Applied Fracture Mechanics, 2012. P. 197 – 228. DOI: 10.5772/52794

11. Kablov E. N. Innovative developments of FSUE «VIAM» of the State Scientific Center of the Russian Federation for the implementation of the «Strategic Directions for the Development of Materials and Technologies for Their Processing for the Period until 2030» / Aviats. Mater. Tekhnol. 2015. N 1(34). P. 3 – 33 [in Russian]. DOI: 10.18577/2071-9140-2015-0-1-3-33

12. Kablov E. N. Strategic directions for the development of materials and technologies for their processing for the period up to 2030 / Aviats. Mater. Tekhnol. 2012. N S. P. 7 – 17 [in Russian].

13. Kalashnikov V. S., Reshetilo L. P., Chuchman O. V., Naprienko S. A. Characteristics of strength and endurance of bars and forgings of blades made of serial heat-resistant titanium alloys and a new pseudo- α-class titanium alloy / Tr. VIAM: Élektron. Nauch.-Tekhn. Zh. 2022. N 2 (108). P. 13 – 31 [in Russian]. DOI: 10/18577/2307-6046-2022-0-2-13-31

14. Kablov E. N., Kashapov O. S., Medvedev P. N., Pavlova T. V. Investigation of a two-phase titanium alloy of the Ti-Al-Sn-Zr-Si-β- stabilizers system / Aviats. Mater. Tekhnol. 2020. N 1(58). P. 30 – 37 [in Russian]. DOI: 10.18577/2071-9140-2020-0-1-30-37

15. Gromov V. I., Yakusheva N. A., Vostrikov A. V., Cherkashneva N. N. High-strength structural steels for gas turbine engine shafts (review) / Aviats. Mater. Tekhnol. 2021. N 1. P. 3 – 12 [in Russian]. DOI: 10.18577/2713-0193-2021-0-1-3-12

16. Gorbovets M. A., Khodinev I. A., Ryzhkov P. V. Equipment for low- cycle fatigue testing under a «hard» loading cycle / Tr. VIAM: Élektron. Nauch.-Tekhn. Zh. 2018. N 9. P. 51 – 60 [in Russian]. DOI: 10.18577/2307-6046-2018-0-9-51-60

17. Paris P., Erdogan F. A critical analysis of crack propagation laws / Journal of Basic Engineering (Trans. ASME). 1963. N 12. P. 528 – 534.

18. Gorbovets M. A., Nochovnaya N. A. On the parameters of the Paris equation in testing the fatigue crack growth rate of high-temperature titanium alloys / Tr. VIAM: Élektron. Nauch.-Tekhn. Zh. 2016. N 4. P. 13 – 19 [in Russian]. DOI: 10.18577/2307-6046-2016-0-4-2-2

19. Korsunsky A. V., Dini D. D., Walsh M. J. Fatigue crack growth rate analysis in a titanium alloy / Key Engineering Materials Vols. 2008. Vol. 385 – 387. P. 5 – 8. DOI: 10.4028/www.scientific.net/KEM.385-387.5