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A practical randomization approach of deterministic equation to determine probabilistic fatigue and fracture behaviours based on small experimental data sets

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Abstract

This paper outlines a new technique to address the paucity of data in determining fatigue and fracture performances based on reliability concepts. Two new randomized models of time-dependent processes are presented for estimating the P–a–t and P–S–N curves, by using a randomization approach of deterministic equations, dealing with small sample numbers of data. The confidence level formulations for these curves are also given. The concepts are then applied for the determination of the P–a–t and P–S–N curves. Two sets of fatigue and fracture tests for these curves are conducted to validate the presented method, demonstrating the practical use of the proposed technique.

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Abbreviations

a :

crack length

a 0 :

initial crack size

\(\hat{{a}}\) :

crack size difference

a(t):

crack length dependent on time t

b 1 :

undetermined parameter of linear regression equation

b 2 :

undetermined parameter of linear regression equation

b i :

undetermined material constant

C :

undetermined material constant

c i :

undetermined material constant

D(x):

standard deviation of variable x

E(x):

expected value of variable x

F :

non-negative function

f(x):

probability density function

K :

stress intensity factor

K max :

spectrum peak stress intensity factor

K C :

facture toughness

L :

likelihood function

L xx :

transformation variable of parameter estimation

L yy :

transformation variable of parameter estimation

L xy :

transformation variable of parameter estimation

m :

undetermined material constant

m i :

undetermined material constant

n :

sample size, or undetermined material constant of crack growth law

N :

fatigue life, or cycles of cyclic loading

N p :

safe fatigue life with a p percent reliability level

p :

reliability level

Q :

transformation variable

R :

stress ratio

s :

standard deviation of the sample

S :

fatigue strength, or stress amplitude, or peak stress level in the loading spectrum

S 0 :

undetermined material constant

S min :

minimum stress

S max :

maximum stress

t :

statistical variable, or time, or number of cycles

t 0 :

initial time

t γ :

γ percentile of t-distribution

u p :

standard normal deviator corresponding to the reliability level of p

u γ :

standard normal deviator corresponding to the confidence level of γ

U :

statistical variable

\(\bar{{x}}\) :

sample mean of random variable X

Y :

random variable

Y(a):

stress intensity boundary correction factor

\(\bar{{y}}\) :

sample mean of random variable Y

y p :

percentiles of random variable Y pertinent to reliability level p

y pγ :

one-side low limit of random variable Y corresponding to the probability of survival of p and the confidence level of γ

Z(a):

normal random process dependent on crack size a

α 0 :

plane stress/strain constraint factor

β:

correction coefficient of the standard deviation

γ:

confidence level

ω:

frequency in radians per second

υ:

degrees of freedom

χ 2 :

statistical variable

μ:

population mean value

σ:

standard variation of random variable

σ z :

standard deviation of normal random process Z(a)

τ 0 :

undetermined material constant

\(\Delta K\) :

stress intensity factor range

\(\Delta K_{th} \) :

fracture threshold value

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Authors and Affiliations

  1. Aircraft Department, Beijing University of Aeronautics and Astronautics, Beijing, 100083, P.R. China

    J. J. Xiong

  2. School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    R. A. Shenoi

Authors
  1. J. J. Xiong
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  2. R. A. Shenoi
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Correspondence to R. A. Shenoi.

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Xiong, J.J., Shenoi, R.A. A practical randomization approach of deterministic equation to determine probabilistic fatigue and fracture behaviours based on small experimental data sets. Int J Fract 145, 273–283 (2007). https://doi.org/10.1007/s10704-007-9116-z

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  • DOI: https://doi.org/10.1007/s10704-007-9116-z

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