extreme value theory in fatigue of clean steelsextreme value theory in fatigue of clean steels clive...
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SEAMOCS Oslo 24/10/081
Extreme Value Theoryin Fatigue of Clean Steels
Clive AndersonUniversity of Sheffield, UK
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Metal Fatigue• repeated stress,
• deterioration, failure
• safety and design issues
The Context
Approaches to Studying FatiguePhenomenological – 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
For ,
Constant amplitude cyclic loading 2σ
Fatigue limit σw
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2 Inclusions in Steel & the Rating Problem
inclusions
• propagation of micro-cracks → fatigue failure
• cracks very often originate at 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 adesignated volume
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Can measure sizes S of sections cut by a plane surface
butnot routinely observable
Inference problem: how use data on S to estimate extremes of V?
3 Extreme Value Theory & Stereology
Rating problem: classify steels in terms of
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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
Data: surface diameters S > v0 in knownarea
– a Marked Poisson Process Model
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where GPD
Stereology
For spherical inclusions with centres at points of a Poisson process
Wicksell 1925Thus
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A missing data problem
If V1, …, Vn had been observed, inference would be simple.
Inference: hierarchical model
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MCMCSample repeatedly from completeconditional distributions of unknowns:
expected no. si
a b
n
v1, v2, … , vn
s1, s2, … , sn
prior parameters
unknowns
unknowns
where eg
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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Given the parameters , the distribution of is Generalized Extreme Value.
Predictive distribution of
10 20 30 40 50
0.00
0.05
0.10
0.15
0.20
m
pred
ictiv
e pr
ob d
ensi
tyT7341: predictive pdfof for C = 100
Predictive distribution of = largest V in volume C
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Sensitivity of Inferences to Sphericity?
Generalized Model:• inclusions of same 3-d shape but different sizes,• random uniform orientation , in principle • sizes Generalized Pareto,• centres in homogeneous Poisson process
ThenE( no. inclusions of size , orientation
intersecting plane in shape of size )
for a function depending on the shape.
E( no. inclusions of size intersectingplane 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
In most metal components internal stresses are non-uniform
-2.5-1.5
-0.50.5
1.52.5
-3-2
-10.0
12
30
100
200
300
400
500
600
700
800
Prin
cipa
l stre
ss, M
Pa
X/hole radius
Y/hole radius
Stress in thin plate with hole, under tension
Component fails if a large inclusionoccurs at a point of high stress amplitude
Failure probability under marked Poisson model?
from stress field inferred from measurements
100mm
5mm
50mm 2mm
Reason for interest in inclusions: design of safe steel components
ie
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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 acomponent of volume C
mean no. of inclusions in volume C
proportion experiencing unbearable stress
consider inclusions of size :
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