extreme value theory in metal fatigue - a selective review clive anderson university of sheffield
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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