PSI - Issue 7
387 5
Hans-Jakob Schindler / Procedia Structural Integrity 7 (2017) 383–390 H.-J. Schindler / Structural Integr ty P o edi 00 (2017) 000–000
Fig. 2: Schematic sketch of the plastic zones in the vicinity of the crack tip.
The values of δ i and δ max are obtained from the following basic relations in LEFM:
2 max
2
E m K
E m K
;
(9)
i
=
δ
i δ
=
max
⋅ ⋅
σ
⋅ ⋅
σ
f
f
With regard to the subsequent simplified analysis, it is suitable to distinguish between R < 0.7, where crack closure is likely to occur and to affect the effective range of SIF, and R > 0.7, where this phenomenon can be ruled out for theoretical reasons. Consider first the latter case, which is simpler. In the absence of crack closure, which is guaranteed for R > 0.7, ∆δ due to unloading from K max to K min = K max - ∆ K is approximately
2
K
⋅ ⋅ ∆
for R >0.7
(10)
δ ∆ =
(2 ) σ
E m
f
since the unloading from tensile to compressive plastic yielding is associated with a doubled elastic range. Inserting (9) and (10) in (8) yields
2 2 / +
q
dN da
K
∆
(11)
=
1
1 ⋅ −
2 max
q
K
1
2
2 2
8
2
q
q
m
K
⋅ ⋅ π
σ
⋅
⋅
⋅
f
i
2
K
i
Obviously, eq. (11) has the form of Paris’ law (2), extended by the term in brackets to include the transition to tearing crack extension as K max approaches K i . In the Paris-regime, i.e. for K max << K i , q is readily identified to be q = 2/(n-2). Furthermore, in order to simplify the expression, m = 2.5 can be inserted, since in the CPZ in the lower Paris’ regime plane strain usually prevails, so (15) leads to
2
n
− 2
2
n
−
⋅ ∆ 2
⋅ − 1
2 max
K
2 σ
dN da
for R >0.7
(12)
n
K
=
2
2
2
n
157
−
K
K
⋅
⋅
⋅
f
i
i
In comparison with experimental da/dN curves, (12) shows surprisingly good agreement for various metals, which confirms the used models and the simplifying assumptions. For R < 0.7, certain effects of plasticity-induced crack closure are likely to affect the fatigue process in the CPZ. In particular, the simple relation (10) no longer holds, since local contact between the crack faces adjacent to the CPZ reduces the effective ∆ K. The more general relation
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