Extreme Value Theory
in Fatigue of Clean Steels
Clive Anderson
University of Sheffield, UK
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The Context
Metal Fatigue
• repeated stress,
• deterioration, failure
• safety and design issues
Approaches to Studying Fatigue
Phenomenological – ie empirical testing and
prediction
Micro-structural, micro-mechanical
– theories of crack initiation
and growth
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Outline
1. Background: the Fatigue Limit
2. Inclusions and the Rating Problem
3. Extreme Value Theory & Stereology
4. Design
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1 Background: the fatigue limit
Constant amplitude cyclic loading
2σ
Fatigue limit σw
For
,
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2 Inclusions in Steel & the Rating Problem
• propagation of micro-cracks → fatigue failure
• cracks very often originate at inclusions
inclusions
Rating Problem: classify steel quality in relation to inclusion content
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Murakami’s root area max relationship between
inclusion size and fatigue limit:
in plane perpendicular to
greatest stress
Rating Problem: classify steel in relation to size of largest inclusion in a
designated volume
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3 Extreme Value Theory & Stereology
Rating problem: classify steels in terms of
but
not routinely observable
Can measure sizes S of
sections cut by a plane surface
Inference problem: how use data on S to estimate extremes of V?
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Models for Sizes of Large Inclusions
Initial Model:
• spherical particles
• diameters V distributed as Generalized Pareto above a threshold v0
• centres form a homogeneous Poisson process, mean rate for those
with V > v0 equal to λ0
– a Marked Poisson Process Model
Data: surface diameters S > v0 in known
area
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Stereology
For spherical inclusions with centres at points of a Poisson process
Wicksell 1925
Thus
where
GPD
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Inference: hierarchical model
A missing data problem
If V1, …, Vn had been observed, inference would be simple.
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MCMC
Sample repeatedly from complete
conditional distributions of unknowns:
a
prior parameters
b
unknowns
n
v1, v2, …
, vn
unknowns
s1, s2, …
where eg
, sn
expected no. si
from Wicksell
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Inferences
•
posterior dists of parameters
• posterior distributions of derived quantities
• predictive distributions for further observations
Example: from 112 measurements on clean bearing steel T7341
T7341: posterior pdf of ξ
T7341: posterior pdf of σ and ξ
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Predictive distribution of
= largest V in volume C
Given the parameters
Generalized Extreme Value.
, the distribution of
is
0.15
0.10
0.05
0.00
T7341: predictive pdf
of
for C = 100
predictive prob density
0.20
Predictive distribution of
10
20
30
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40
50
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Sensitivity of Inferences to Sphericity?
Generalized Model:
• inclusions of same 3-d shape but different sizes,
• random uniform orientation
• sizes
, in principle
Generalized Pareto,
• centres in homogeneous Poisson process
Then
E( no. inclusions of size , orientation
intersecting plane in shape of size
for a function
)
depending on the shape.
E( no. inclusions of size intersecting
plane in shape of size
)
where
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Titanium Inclusions
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Predictive Distributions for Max Inclusion MC in Volume C = 100
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4. Use in Design
Reason for interest in inclusions: design of safe steel components
In most metal components internal stresses
are non-uniform
800
600
Component fails if a large inclusion
occurs at a point of high stress amplitude
500
400
300
200
100
-2.5
-1.5
-0.5
ie
X/hole radius
Principal stress, MPa
700
0
0.5
1.5
2.5
-2
-3
-1
0.0
1
3
2
Y/hole radius
Stress in thin plate with hole, under tension
from stress field
inferred from measurements
5mm
100mm
Failure probability under marked Poisson model?
50mm
2mm
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• Under the marked Poisson model:
inclusions at which local stress is too great to bear
≡ thinned (inhomogeneous) Poisson process
If no. of such = N, then
Pr( component fails) = Pr( N > 0)
= 1 – exp( - E(N))
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• Expected no., E(N), of inclusions causing failure in a
component of volume C
consider inclusions of size
mean no. of inclusions in
volume C
:
proportion experiencing unbearable
stress
Over all sizes
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from Generalized Pareto model
from stress distribution and
size – fatigue limit relationship
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Effect of
• modifying the design
• improving cleanness of steel
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