Crack Paths 2009

2 1

K G eff

= Δ Δ

⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+ R a R a

(45)

β π

a

The asymptotic interpolation between these solutions for large and small cracks on one

side ofthe inclusion is:

Δ R π a FxΔ,γ( )

eff

(46)

KΔ = G

where: FΔx,γ()=A+Bx1+x⎛⎝⎜⎞⎠⎟+Cx1+x⎛⎝⎜⎞⎠⎟

2 +Dx1+x⎛⎝⎜⎞⎠⎟

3

(47)

with: Aγ()=2πf0 , Bγ()=−8f1 , Cγ()=−6πf0+16f1 , Dγ()=4πf0−8f1 (48)

(49)

and:

)1/(122.010γ++=f ,

)1/(073.011γ++=f

Finally, these results, from Eqs. (37) through (49), may be combined to get the total

(50)

stress intensity factor from: K total

= Kσ + K Δ

The intended application of this work is to analyze the initiation and growth of cracks in

gigacycle fatigue. For that purpose Eq. (50) putting in the maximumstress,σ , to get

the maximumstress intensity factor. For the range of the stress intensity factor, since

the K Δ imposes a constant stress intensity factor, the range should be computed from

max, min,/ total total K K is

the Kσ only by Eq. (51). But, Eq. (50) shows that the R ratio

min, / σ σ K K

different from

due to the cyclic loading.

max,

(51)

ΔK = ΔKσ

total

C O N C L U S I O N

The local elastic stresses and the theoretical stress concentration factors near inclusions

mismatches in elastic properties and sizes, and included in a matrix under uniaxial

loading or external hydrostatic tension were computed. The stress intensity factors for

cracks initiating from them was approximated too by some elastic stress formulas. It is

hoped that convenient availability of such formulas may enhance an understanding of

gigacycle fatigue initiation and crack growth.

Acknowledgements

The authors acknowledge Arts et Metiers ParisTech and Foundation Arts et Metiers for

the financial support of Paul C. Paris’ stay at LAMEFIPin 2009. The encouragement of

Prof. Ivan Iordanoff, Director of LAMEFIP,is also acknowledged with thanks

R E F E R E N C E S

1. Peterson R.E. (1974) Stress concentration factors, Wiley-Interscience, New-York.

2. Tada H., Paris P.C. and Irwin G.R. (2000) The Stress Analysis of Cracks Handbook, 3rd edition, A S M EPress.

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