Extreme Value Theory in Metal Fatigue

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

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The Context

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

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Aims

• Understanding
• Prediction

Approaches

1. Phenomenological ie empirical testing and
   prediction
2. Micro-structural, micro-mechanical
   theories of crack
   initiation
   and growth

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1.1 Testing the idealized S-N (Wohler) Curve
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Example S-N Measurements for a Cr-Mo Steel
Variability in properties suggesting a
stochastic formulation
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Some stochastic formulations
(Murakami)
whence extreme value distribution for
N(s) no. cycles to failure at stress s gt sw
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Some Inference Issues
de Maré, Svensson, Loren, Meeker
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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
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Rainflow filtering
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Damage Accumulation Models
(Palmgren-Miner rule)
Knowledge of load process and of S - N relation
in principle allow prediction of life
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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?

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• 2.1 Inclusions in Steel

• propagation of micro-cracks ? fatigue failure
• cracks very often originate at inclusions

inclusions
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Murakamis root area max relationship between
inclusion size and fatigue limit
in plane perpendicular to greatest stress
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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 gt v0 in known area
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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
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Predictive Distributions for Max Inclusion MC in
Volume C 100
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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

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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
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Illustrative only other effects operating

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

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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

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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,
22952304 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, 205220 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, 14551468 Svensson, T de
Mare, J (1999) Random features of the fatigue
limit. Extremes 2, 149-164