PSI - Issue 4

Hans-Jakob Schindler / Procedia Structural Integrity 4 (2017) 48–55 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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3

the crack faces. If K min + K Irs < K rem , which in general holds for rotating bending in a railway axle, then K min + K Irs < K rem has to be replaced by K rem , so (2) is modified to

(3)

rem K K K K R K       max max ; rem Irs K K

rs

K rem is a positive constant that depends on the roughness of the crack surface and the thickness of corrosion products adhering to it. K rem is hardly predictable on pure theoretical grounds, so it needs to be determined by realistic component testing. For the sake of simplicity and conservatism we assume in the following

K rem = 0

(4)

With (3) and (4), (1) leads to





dN da





(5)

n

(    K R n th

( )     C R K a K a ( ) ( 0)

0)

Irs

max

From comparison of (5) with (1), the large effect of the residual stress on the crack growth rate is evident: It results in an increase of the effective range of K I by K Irs .

3. Determination of K Irs (a) by the Cut-Compliance Method In general, the SIF due to residual stresses can be determined by

a

rs    ( ) ( , ) ( ) 0 

(6)

K a Irs

x h x a dx 

where h(x,a) denotes the weight function for a surface crack and  rs (x) the original residual stress distribution along the crack-path (Wu and Carlsson (1991)). To apply (6), the stress-profile  rs (x) needs to be known. A suitable method to measure residual stress profiles is the Cut-Compliance Method (CC-Method) as proposed by Schindler et al. (1997) and Schindler and Bertschinger (1997). It requires a cut (actual length a, width e; see Fig. 1) to be introduced along the line of interest (x), and measurement of the strain  M (a) at suitable locations as a function of cut depth a.

da d a M 

( )

( ) '

(7)

Z a K a E Irs ( ) 



In (7), E’ denotes the generalized Young’s modulus,  M (a) the strain measured at M1 and/or at M2 (see Fig. 1), and Z(a) the influence-function of the corresponding measurement point. From K Irs (a) as obtained in this way, the residual stress distribution  rs (x) can be obtained by inversion of eq. (6). In order to calculate the fatigue crack growth by means of eq. (5), there is no need to determine  rs (x) by (6); K Irs as obtained from eq. (7) is sufficient.