Extreme Value Theory in Metal Fatigue a Selective Review Clive Anderson University of Sheffield

1. Extreme Value Theory in Metal Fatigue • a Selective Review • Clive Anderson • University of Sheffield

2. The Context • Metal Fatigue • repeated stress, • deterioration, failure • safety and design issues

3. Aims • Understanding • Prediction Approaches • Phenomenological – ie empirical testing and prediction • Micro-structural, micro-mechanical – theories of crack initiation and growth

4. 2σ Constant amplitude cyclic loading Fatigue limit sw For , 1.1 Testing: the idealized S-N (Wohler)Curve

5. Example: S-N Measurements for a Cr-Mo Steel Variability in properties – suggesting a stochastic formulation

6. given often taken linear in giving approx, some Some stochastic formulations: (Murakami) whence extreme value distribution for N(σ) = no. cycles to failure at stress σ > σw

7. precision under censoring, discrimination between models • design in testing, choice of test , ancillarity • hierarchical modelling, simulation-based methods Some Inference Issues: de Maré, Svensson, Loren, Meeker …

8. stress 1.2 Prediction of fatigue life In practice - variable loading Empirical fact: local max and min matter, but not small oscillations or exact load path. Counting or filtering methods: eg rainflow filtering, counts of interval crossings,… functions of local extremes to give a sequence of cycles of equivalent stress amplitudes

9. stress th rainflow cycle stress amplitude Rainflow filtering

10. eg if damage additive and one cycle at amplitude uses up of life, total damage by time Fatigue life = time when reaches 1 Damage Accumulation Models (Palmgren-Miner rule) Knowledge of load process and of S - N relation in principle allow prediction of life

11. Issues: • implementationMarkov models for turning points, approximations for transformed Gaussian processes, extensions to switching processes WAFO – software for doing theseLindgren, Rychlik, Johannesson, Leadbetter…. • materials with memory damage not additive, simulation methods?

12. propagation of micro-cracks → fatigue failure • cracks very often originate at inclusions inclusions 2.1 Inclusions in Steel

13. Murakami’s root area max relationshipbetween inclusion size and fatigue limit: in plane perpendicular to greatest stress

14. not routinely observable Can measure sizes S of sections cut by a plane surface • Model: • inclusions of same 3-d shape, but different sizes • random uniform orientation • sizes Generalized Pareto distributed over a threshold • centres in homogeneous Poisson process Data: surface areas > v0 in known area

15. for some function Inference for : Murakami, Beretta, Takahashi, Drees, Reiss, Anderson, Coles, de Maré, Rootzén… • stereology • EV distributions • hierarchical modelling • MCMC Results depend on shape through a function B

16. Predictive Distributions for Max Inclusion MCin Volume C = 100

17. Stress in thin plate with hole, under tension Application: Failure Probability & Component Design In most metal components internal stresses are non-uniform Component fails if at any inclusion from stress field inferred from measurements • If inclusion positions are random, get simple expression for failure probability, giving a design tool to explore effect of: • changes to geometry • changes in quality of steel

18. Tundish inclusion size pdfon exit prob. inclusion does not reach slag layer inclusion size pdf on entry Simple laminar flow: So 2.2 Genesis of Large Inclusions Modelling of the processes of production and refining shouldgive information about the sizes of inclusions Example: bearing steel production – flow through tundish Mechanism: flotation according to Stokes Law ie GPD with  = -3/4 almost irrespective of entry pdf

19. Illustrative only: other effects operating • complex flow patterns • agglomeration • ladle refining & vacuum de-gassing • chemical changes

20. Approach for complex problems: • model initial positions and sizes of inclusions by a marked point process • treat the refining process in terms of a thinning of the point process • use computational fluid dynamics & thermodynamics software – that can compute paths/evolution of particles – to calculate (eg by Monte Carlo) intensity in the thinned processand hence size-distribution of large particles • combine with sizes measured on finished samples of the steel eg via MCMC

21. Some references: www.shef.ac.uk/~st1cwa Anderson, C & Coles, S (2002)The largest inclusions in a piece of steel. Extremes 5, 237-252 Anderson, C, de Mare, J & Rootzen, H. (2005) Methods for estimating the sizes of large inclusions in clean steels, Acta Materialia 53, 2295—2304 Beretta, S & Murakami, Y (1998) Statistical analysis of defects for fatigue strength prediction and quality control of materials. FFEMS 21, 1049--1065 Brodtkob, P, Johannesson, P, Lindgren, G, Rychlik, I, Ryden, J, Sjo, E & Skold, M (2000) WAFO Manual, Lund Drees, H & Reiss, R (1992) Tail behaviour in Wicksell's corpuscle problem. In ‘Prob. & Applics: Essays in Memory of Mogyorodi’ (eds. J Galambos & I Katai) Kluwer, 205—220 Johannesson, P (1998) Rainflow cycles for switching processes with Markov structure. Prob. Eng. & Inf. Sci. 12, 143-175 Loren, S (2003) Fatigue limit estimated using finite lives. FFEMS 26, 757-766 Murakami, Y (2002) Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Elsevier. Rychlik, I, Johannesson, P & Leadbetter, M (1997) Modelling and statistical analysis of ocean wave data using transformed Gaussian processes. Marine Struct. 10, 13-47 Shi, G, Atkinson, H, Sellars, C & Anderson, C (1999) Applic of the Gen Pareto dist to the estimation of the size of the maximum inclusion in clean steels. Acta Mat 47, 1455—1468 Svensson, T & de Mare, J (1999) Random features of the fatigue limit. Extremes 2, 149-164