PSI - Issue 7

Hans-Jakob Schindler / Procedia Structural Integrity 7 (2017) 383–390 H.-J. Schindler / Structural Integrity Procedia 00 (2017) 000–000

389

7

2

n

− 2

2

n

−

  

 ⋅ ∆   2

R R (1 ) 1 − − 2

⋅ 0.088 σ 2

dN da dN da

for 0 < R < 0.7

(17a)

n

C

K

=

⋅

⋅

nd

2

2

n Ic

157

−

K

⋅

f

2

n

− 2

2

for R > 0.7

(17b)

n

C

K

=

⋅

⋅ ∆

nd

2

2

n Ic

157

−

K

⋅

⋅

σ

f

For R < 0, as a first approximation K min can be replaced by zero due to remote crack closure, which leads to R = 0 and ∆ K = K max in eq. (17a), thus

2

n

2 −

0.088

dN da

for R < 0

(17c)

n

C

K

=

⋅

⋅ ∆

max

nd

2

2

n Ic

157

−

K

⋅

⋅

σ

f

Approximately, eqs. (17a-c) hold for ∆ K th < ∆ K < K Ic · (1-R). It is expected that C nd is much less material-dependent than C P . Preliminary comparisons with experimental da/dN-data indicate that C nd ≈ 0.5, depending only little on n and material. As a matter of experience, n can be assumed to be 3. Since no assumptions have been made in the derivation concerning the material, it is expected to be applicable to any elastic-plastic metal. In the near-threshold range, eqs. (17a-c) can be refined by replacing ∆ K n by ( ∆ K n - ∆ K th n (R)).

Fig. 3: Comparison of S(R) according to eq. (15) (“analytical”) with eq. (3) (“ASME”)

5. Discussion and conclusions Eqs. (17a-c) represent a generalization and extension of Paris’ law, including the stress ratio R as a parameter. However, unlike most of the many extensions and modifications of Paris’ law that are proposed in the literature, the presented ones don’t contain additional open parameters. In contrary: Since all the above relations are dimensionally correct, the calibration factor C nd is non-dimensional, and varies only in a relatively narrow range of about 0.2 to 1, according to some first comparisons with experimental data. It is expected that C nd is a constant within a certain class of materials. For these reasons, determination of C nd is much less demanding than C P with regard to the quantity and quality of experimental data. Moreover, the presented analysis provides some theoretical support to the essentially empirical “Paris’ law” and some insight in the physics behind FCG. It reveals that the flow stress and the upper-shelf fracture toughness are the main influencing parameters of C P , and it enables the effect of changes of these properties on FCG to be predicted. Vice-versa, the good agreements with Paris’ law and with experimental data confirm the assumptions made in the