Extreme Value Theory in Fatigue of Clean SteelsExtreme Value Theory in Fatigue of Clean Steels Clive...

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Extreme Value Theoryin Fatigue of Clean Steels

Clive AndersonUniversity of Sheffield, UK


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


Outline

1. Background: the Fatigue Limit

2. Inclusions and the Rating Problem

3. Extreme Value Theory & Stereology

4. Design


1 Background: the fatigue limit

For ,

Constant amplitude cyclic loading 2σ

Fatigue limit σw


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


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


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


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


where GPD

Stereology

For spherical inclusions with centres at points of a Poisson process

Wicksell 1925Thus


A missing data problem

If V1, …, Vn had been observed, inference would be simple.

Inference: hierarchical model


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


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


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


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


Titanium Inclusions


Predictive Distributions for Max Inclusion MC in Volume C = 100


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


• 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))


• 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


from Generalized Pareto model from stress distribution and size – fatigue limit relationship


Effect of • modifying the design• improving cleanness of steel

Extreme Value Theory in Fatigue of Clean SteelsExtreme Value Theory in Fatigue of Clean Steels Clive...

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