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

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

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

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

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

8. 1.2 Prediction of fatigue life
stress
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. Rainflow filtering
stress
th rainflow cycle
stress amplitude

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

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

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

13. Murakami’s root area max relationship between
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. hierarchical modelling MCMC
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
for some function

16. Predictive Distributions for Max Inclusion MC in Volume C = 100

17. Application: Failure Probability & Component Design
In most metal components internal stresses are non-uniform
Stress in thin plate with hole, under tension
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. 2.2 Genesis of Large Inclusions
Modelling of the processes of production and refining should give information about the sizes of inclusions
Example: bearing steel production – flow through tundish
Tundish
Mechanism: flotation according to Stokes Law
inclusion size pdf on exit
inclusion size pdf on entry
prob. inclusion does not reach slag layer
Simple laminar flow: So
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 process and 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,
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,
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,
Loren, S (2003) Fatigue limit estimated using finite lives. FFEMS 26,
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,