Download - 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
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Metal Fatigue
• repeated stress,
• deterioration, failure
• safety and design issues
The Context
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Aims
Approaches
• Understanding
• Prediction
1. Phenomenological – ie empirical testing and prediction
2. Micro-structural, micro-mechanical – theories of crack initiation and growth
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1.1 Testing: the idealized S-N (Wohler) Curve
Fatigue limit w
For ,
Constant amplitude cyclic loading
2σ
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Example: S-N Measurements for a Cr-Mo Steel
Variability in properties – suggesting a stochastic formulation
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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
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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 …
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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
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stre
ss
th rainflow cycle
Rainflow filtering
stress amplitude
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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
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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?
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2.1 Inclusions in Steel
inclusions
• propagation of micro-cracks → fatigue failure
• cracks very often originate at inclusions
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Murakami’s root area max relationship between inclusion size and fatigue limit:
in plane perpendicular to greatest stress
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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
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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…
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Predictive Distributions for Max Inclusion MC in Volume C = 100
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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
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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
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Illustrative only: other effects operating
• complex flow patterns
• agglomeration
• ladle refining & vacuum de-gassing
• chemical changes
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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
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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