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Extreme Value Theory in Metal Fatigue a Selective Review Clive Anderson University of Sheffield. The Context. Metal Fatigue repeated stress, deterioration, failure safety and design issues. Aims . Understanding Prediction. Approaches.

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Extreme Value Theory in Metal Fatigue

  • a Selective Review
  • Clive Anderson
  • University of Sheffield

The Context

  • Metal Fatigue
    • repeated stress,
    • deterioration, failure
    • safety and design issues


  • Understanding
  • Prediction


  • Phenomenological – ie empirical testing and prediction
  • Micro-structural, micro-mechanical – theories of crack initiation and growth

Constant amplitude cyclic loading

Fatigue limit sw

For ,

1.1 Testing: the idealized S-N (Wohler)Curve


Example: S-N Measurements for a Cr-Mo Steel

Variability in properties – suggesting a stochastic formulation



often taken linear in


approx, some

Some stochastic formulations:


whence extreme value distribution for

N(σ) = no. cycles to failure at stress σ > σw


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 …



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



th rainflow cycle

stress amplitude

Rainflow filtering


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



  • 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?

propagation of micro-cracks → fatigue failure

  • cracks very often originate at inclusions


2.1 Inclusions in Steel


Murakami’s root area max relationshipbetween

inclusion size and fatigue limit:

in plane perpendicular to greatest stress


not routinely


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


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


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


inclusion size pdfon exit

prob. inclusion does not reach slag layer

inclusion size pdf on entry

Simple laminar flow:


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


Illustrative only: other effects operating

  • complex flow patterns
  • agglomeration
  • ladle refining & vacuum de-gassing
  • chemical changes

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

Some references:

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