Fatigue Crack Paths 2003

where G is the shear modulus, ν is the Poisson’s ratio and L is the distance from the

crack tip. The variation of KI and KII for leading crack tip with various positions of the

wheel is shown in Fig. 5. The local KI and KII for a prospective α kinking angle are

obtained by [14]:

) (

(3)

K

α =

II K C K C 12 11 + I

I

) (

(4)

K

α =

II K C K C 22 21 + I

II

where KI(α) and KII(α) are the local stress intensity factors at the tip of the kink and KI

and KII are the stress intensity factors for the main crack, obtained by F E Manalysis and

Eqs 1 and 2. The coefficients Cijare given by:

⎟⎠⎞⎜⎝⎛ = 2 3 4 1 2 4 3 11 α α cos cos C +⎟⎠⎞⎜⎝⎛

(5a)

sin

sin

(5b)

⎢⎣⎡ ⎟⎠⎞⎜⎝⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 23 2 4 3 12 α α C

⎥⎦⎤

sin

sin

⎢⎣⎡ 23 2 4 1 α α ⎟⎠⎞⎜⎝⎛ +⎟⎠⎞⎜⎝⎛

(5c)

C

=

⎥⎦⎤

21

⎟⎠⎞⎜⎝⎛ = 2 3 4 3 2 4 1 22 α α cos cos C +⎟⎠⎞⎜⎝⎛

(5d)

For each angle and for each simulated step (in Fig. 5 is only represented the case of

α=0, that means no kinking), two main values are obtained: ΔΚII and ΔKI. For the last

one, ΔKI, crack closure phenomena is neglected.

Crack GrowthDirection

Several F E Msimulations were carried out for different steps of crack propagation.

Figure 6 shows, for the rail failure represented on Fig. 1a, the three simulated crack tip

positions. Figures 7 to 9 show the results for the leading tip. In all cases, αrepresents

the angle between possible kinking and the present direction of the crack: α=0means

non deviation (Fig. 6). At the first and second step, it could be appreciate (Figs 7 and 8)

that propagation is associated with a maximumlevel of ΔKII together with a small KI. In

particular, the actual direction of propagation corresponds to a ΔKII value near to the

maximumone which is “helped” by a small KI. Figure 9, in turn, shows the results for

the stress intensity factors when the crack is just before kinking. As it can be observed,

the real angle of propagation corresponds to maximummode I, so it is clear that kinking