PSI - Issue 7

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

386

4

1 reveals that (4) overestimates the R-dependence of C p for R < f, whereas eq. (5) rises too steep for larger R. The latter effect was realized by Huang and Moan (2007) and corrected by a modification of (5) in the range 0.5 < R < 1. For R < f, there is crack closure. The large deviation of eq. (4) from eq. (3) in this range indicates that crack closure does not explain satisfyingly the effect of R on C P . Fig. 1 rather supports the skepticism of Vasudevan (1994), Kujawski (2001), Huang and Moan (2007) concerning crack closure as the main cause of the R-dependence of C p . Indeed, K* seems to work better, provided experimental data are available to determine α properly. However, a convincing explanation of the physical meaning of K* or a plausible description of mechanisms that causes the R-dependence without taking credit from crack closure is not given by the aforementioned authors who advocate K*. It is easy to show theoretically that if crack closure was the main reason for R-dependence of C P , then da/dN should correlate with K max . In reverse, however, da/dN-data that correlate with K max are not necessarily associated with crack closure, as noticed by Schindler and Leinenbach (2010 and 2012) in the case of FCG in a steel brazing. Although the experimental da/dN data for 0.1 < R < 0.7 collapsed almost perfectly into one scatter band if plotted as a function of K max , direct measurement of K op by the cut-compliance method revealed that practically no crack closure was present (Schindler and Leinenbach (2010). This phenomenon could be explained theoretically by assuming that the cyclic range of crack-tip opening displacement (CTOD), ∆δ = δ max – δ min , represents the main driving force for FCG, rather than ∆ K (Schindler and Leinenbach (2012)). Based on this hypothesis, they could derive da/dN-curves in a brazed layer analytically in good agreement with the experimental data. In particular, the analytical curves exhibited the experimentally observed R-dependence of da/dN, without taking credit from crack closure. In the following, a similar model is chosen to predict FCG in homogeneous elastic-plastic material. 3. Analytical considerations FCG results from the fatigue damage that is accumulated in the cyclic plastic zone (CPZ) where the material is subjected to reversed plastic yielding (Fig. 2). From fundamental relations of LEFM, the length of this zone is known to be The cyclic plastic strain at the crack tip that causes damage by local low-cycle fatigue (LCF) can be assumed to be proportional to the cyclic range of CTOD, ∆δ . Thus, analogously to the well-known Coffin-Mansion relation in LCF, it can be assumed that the number of strain cycles, N f , required to break the material particle at the crack tip is related to the range of CTOD in the following way: max δ δ δ ∆ ⋅ = − i q f N (7) where q represents an open non-dimensional parameter and δ i the CTOD required to initiate ductile tearing. If the crack is in a steady process of FCG, then the material particles on the x-axis (see Fig. 2) are moving through the CPZ, entering it at x = r pR in a virgin state and leaving it broken at x = 0 due to local LCF. Assuming, for the sake of simplicity, that the plastic strain range within the CPZ varies linearly between the maximum at the crack-tip (x=0) and zero at x = r pR and that the damage accumulation is linear, a material particle withstands 2 · N f load cycles, then it breaks. Correspondingly, da/dN is given roughly by 2 2 ) 4 ( π ⋅ f pR m K r σ ⋅ ∆ = (6)

1

r

K ∆ ⋅ ∆ 2

q

dN da

δ

pR = =

(8)

N 2 8

1

2

2

(

max δ δ )

q

m

π ⋅ ⋅

⋅

σ

⋅

−

f

f

i