Assessment of failure probabilities and the allowable size of defects in structural elements using the criteria of fracture mechanics

The possibility of assessing the safe size of the defects of metal continuity using risk criteria are considered. Such defects occur at all stages of the life cycle of structures. Assessment of their hazard and determination of their allowable size becomes important when the defects can lead to brittle or quasi-brittle fracture. In this case, models of linear and nonlinear fracture mechanics are used. In these models, defects are considered as internal elliptical or surface semi-elliptical cracks. The stochastic variety of shapes, sizes, locations, and orientations of defects has a significant effect on the destruction mechanisms. In this regard, the relevant probabilistic problem of assessing the permissible size of defects according to the criteria of the risk of destruction comes to the fore. We consider a general approach to assessing hazard of defects by risk criteria. Two settings of the probabilistic problem of risk assessment based on of one- and two-parameter fracture criteria are presented. The risk function in the form of the probability of fracture according to a given criterion is used as the main calculated characteristic. The expression of the risk function based on one-parameter criteria of fracture mechanics is presented. The main attention is paid to the probabilistic model based on the two-parameter fracture criterion by E. M. Morozov. This criterion provides ample opportunities for analyzing fracture mechanisms with a change in the size of defects. An expression for the risk function based on the family of two-dimensional probability distributions of Lu – Bhattacharya of the Weibull type is obtained. It is shown that the correlations of the fracture mechanisms can significantly affect the values of the fracture probabilities, and, consequently, the allowable size of defects.

1. Anderson T. L. Fracture mechanics. Fundamentals and Application. — CRC Press. 2017, — 661 p.

2. Fracture mechanics. Theory, applications and research / Robertson J. C., Ed. — Nova Science Publishers, 2017. — 124 p.

3. Matvienko Yu. G. Two-parameter fracture mechanics. — Moscow: Fizmatlit, 2020. — 208 p. [in Russian].

4. Moskvichev V. V., Makhutov N. A., Shokin Yu. I., Lepikhin A. M. Applied problems of structural strength and fracture mechanics of technical systems. — Novosibirsk: Nauka, 2021. — 796 p. [in Russian].

5. Arkadov G. V., Getman A. F., Rodionov A. N. Reliability of NPP equipment and pipelines and optimization of their life cycle. — Moscow: Énergoatomizdat, 2010. — 424 p. [in Russian].

6. Lepikhin A. M., Makhutov N. A., Shokin Yu. I. Probabilistic multiscale fracture modelling of heterogeneous materials and structures / Zavod. Lab. Diagn. Mater. 2020. Vol. 86. N 7. P. 45 – 54 [in Russian]. DOI: 10.26896/1028-6861-2020-86-7-45-54

7. Lepikhin A. M., Makhutov N. A., Shokin Yu. I., Yurchenko A. V. The concept of risk analysis of technical systems using digital twins / Vychisl. Tekhnol. 2020. Vol. 25. N 4. P. 99 – 113 [in Russian]. DOI: 10.25743/ICT.2020.25.4.009

8. Bazant Z. P., Le J.-L. Probabilistic mechanics of quasibrittle structures. Strength, lifetime, and size effect. — Cambridge university press, 2017. — 302 p.

9. Cui L., Frenkel I., Lisnianski A. Stochastic models in reliability engineering. — CRC Press, 2021. — 464 p.

10. Morozov E. M., Fridman Ya. B. Crack analysis as method for assessing fracture characteristics / Zavod. Lab. 1966. Vol. 32. N 8. P. 977 – 984 [in Russian].

11. Morozov E. M. Limit analysis for structures with flows / Eng. Fract. Mech. 1974. Vol. 6. N 1. P. 297 – 306.

12. Makhutov N. A. Safety and risk: systems research and development. — Novosibirsk: Nauka, 2017. — 724 p. [in Russian].

13. Hamad H., Bai Y., Ali L. A risk-based inspection planning methodology for integrity management of subsea pipelines / Ships and offshore structures. 2021. Vol. 16. Issue 7. P. 687 – 699. DOI: 10.1080/17445302.2020.1747751

14. Dowling A. R., Townley C. H. A. The effect of defects on structural failure: A two-criteria approach / Int. J. Pressure Vessels Piping. 1975. Vol. 3. Issue 2. P. 77 – 107.

15. Pestrikov V. M., Morozov E. M. Fracture mechanics. — St. Petersburg: TsOP Professiya, 2012. — 552 p. [in Russian].

16. Altmetwally E. M., Muhammed H. Z., El-Sherpieny E. A. Bivariate Weibull distribution: properties and different methods of estimation / Ann. Data Sci. 2020. Vol. 7(1). P. 163 – 193. DOI: 10.1007/s40745-019-00197-5

17. Mondal S., Kundu D. A bivariate inverse Weibull distribution and its application in complimentary risk model / J. Appl. Stat. 2020. Vol. 47. Issue 6. P. 1048 – 1108. DOI: 10.1080/02664763.2019.1669542

18. Kydryavtsev D. A., Lezin I. A. Approximation of two-dimensional distribution laws of dependent random variables / Izv. Samar. NTs RAN. 2014. Vol. 16. N 4(2). P. 322 – 324 [in Russian].

19. Shayanfar M. A., Barkhordari M. A., Roudak M. A. An adaptive importance sampling-based algorithm using the first order method for structural reliability / Int. J. Optim. Civil Eng. 2017. Vol. 7(1). P. 93 – 107.