Extreme Value Theory in Metal Fatigue

- a Selective Review

Clive Anderson

University of Sheffield

Metal Fatigue

• repeated stress,

• deterioration, failure

• safety and design issues

The Context

Aims

Approaches

• Understanding

• Prediction

1. Phenomenological – ie empirical testing and prediction

2. Micro-structural, micro-mechanical – theories of crack initiation and growth

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

Fatigue limit w

For ,

Constant amplitude cyclic loading

2σ

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

Variability in properties – suggesting a stochastic formulation

Some stochastic formulations:

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

whence extreme value distribution for

given

(Murakami)

often taken linear in

giving

approx, some

Some Inference Issues:

• precision under censoring, discrimination between

models

• design in testing, choice of test , ancillarity

• hierarchical modelling, simulation-based methods

de Maré, Svensson, Loren, Meeker …

1.2 Prediction of fatigue life

In practice - variable loading

stre

ss

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

stre

ss

th rainflow cycle

Rainflow filtering

stress amplitude

Damage Accumulation Models

eg if damage additive and one cycle at amplitude uses up of life,

total damage by time

(Palmgren-Miner rule)

Fatigue life = time when reaches 1

Knowledge of load process and of S - N relation in principle allow prediction of life

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?

2.1 Inclusions in Steel

inclusions

• propagation of micro-cracks → fatigue failure

• cracks very often originate at inclusions

Murakami’s root area max relationship between inclusion size and fatigue limit:

in plane perpendicular to greatest stress

Can measure sizes S of sections cut by a plane surface

not routinely observable

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

Inference for :• stereology• EV distributions• hierarchical modelling• MCMC

for some function

Results depend on shape through a function B

Murakami, Beretta, Takahashi,Drees, Reiss, Anderson, Coles, de Maré, Rootzén…

Predictive Distributions for Max Inclusion MC in Volume C = 100

Application: Failure Probability & Component Design

In most metal components internal stresses are non-uniform

-2.5-1.5

-0.50.5

1.5

2.5

-3-2

-10.0

12

3

0

100

200

300

400

500

600

700

800

Prin

cipa

l str

ess,

MP

a

X/hole radius

Y/hole radius

Stress in thin plate with hole, under tension

Component fails if at any inclusion

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

from stress field inferred from measurements

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 Tundish

Simple laminar flow:

ie GPD with = -3/4 almost irrespective of entry pdf

inclusion size pdfon exit

inclusion size pdf on entry

prob. inclusion does not reach slag layer

So

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

www.shef.ac.uk/~st1cwa